Function Notation with an Equation
Summary
TLDRThis video script delves into the application of function notation through equations, providing a clear explanation of how to interpret and calculate outputs for given inputs. It emphasizes the importance of understanding the function's formula and order of operations. The script walks through several examples, including handling absolute values and constants, to demonstrate how to apply function notation in algebraic expressions. It aims to clarify common mistakes and misconceptions, encouraging viewers to practice and become proficient in using function notation.
Takeaways
- 📚 Function notation is a common mathematical tool used to represent the relationship between inputs and outputs in equations.
- 🔍 The function notation typically includes a function name (like F, G, or H) followed by the input variable (X) and an equals sign, indicating the output (Y).
- 📐 The formula for a function provides a generic way to calculate outputs for any given input, unlike visual representations which show specific mappings.
- 🔑 The variable part of the equation is the input (X), while the function name (like 'f') remains constant and represents the operation being performed.
- 🔄 When using a function formula, you substitute the variable (X) with the desired input value to calculate the corresponding output.
- 🧩 Understanding the order of operations is crucial when substituting inputs into a function formula to ensure correct calculations.
- 📘 The script provides examples of how to calculate outputs for given inputs using function formulas, emphasizing the importance of following the correct mathematical procedures.
- 📉 The concept of absolute value is introduced in the script, which is denoted by vertical bars and requires special handling when substituted into a function.
- 🔢 Functions can be simple or complex, and the script demonstrates how to handle both, including polynomials and functions with absolute values.
- 📝 The script also covers how to input expressions (like X + 1) into functions, resulting in outputs that may involve variables rather than specific numbers.
- 🔄 The process of substituting inputs into functions and simplifying expressions is a fundamental skill in algebra, which the script aims to reinforce.
Q & A
What is the basic concept of function notation?
-Function notation is a way to represent a function where you write the function name, followed by the input variable in parentheses, and then an equals sign and the expression that defines the output.
How is a function defined using an equation?
-A function is defined using an equation by stating the function name, followed by the input variable, an equals sign, and a formula that calculates the output for any given input.
What does 'f(x) = 2(x^2 - 3) / (x - 4)' represent in function notation?
-This represents a function named 'f' that takes an input 'x' and outputs the result of the formula '2(x^2 - 3) / (x - 4)'.
What is the significance of the order of operations in calculating the output of a function?
-The order of operations is crucial in ensuring that the correct output is calculated. It dictates the sequence in which operations within the function's formula should be performed, such as parentheses, exponents, multiplication/division, and addition/subtraction.
How do you calculate the output of the function f(x) when x is 2?
-To calculate the output when x is 2, you replace every instance of 'x' in the function's formula with 2, then follow the order of operations to compute the result, which in this case is -5/2 or negative five-halves.
What does the vertical bar symbol '|' represent in mathematics?
-The vertical bar symbol '|' represents the absolute value in mathematics, which means the non-negative value of whatever is inside the bars, regardless of its original sign.
How does the function G(x) handle the absolute value in its formula?
-The function G(x) first calculates the value inside the absolute value bars, then takes the absolute value of that result, and finally multiplies it by -2 and adds 4 to get the output.
What is the output of the function H(x) for any input x?
-The output of the function H(x) is always -17, as the function is defined to output this value regardless of the input x, because x does not appear in the formula defining H(x).
How can you input an expression like 'x + 1' into a function?
-To input an expression like 'x + 1' into a function, you replace every 'x' in the function's formula with 'x + 1' and then simplify the resulting expression to find the output in terms of x.
Why is it important to understand the definition of a function when using function notation?
-Understanding the definition of a function is important because it provides the specific formula or rule that determines how the function will transform an input into an output, which is essential for accurate calculations.
Can you provide an example of how to calculate the output of a function with a variable input like 'x + 1'?
-Sure, using the function f(x) = 3x^2 - 2x + 1, if the input is 'x + 1', you replace 'x' with 'x + 1' to get f(x + 1) = 3(x + 1)^2 - 2(x + 1) + 1, then simplify to get 3x^2 + 4x + 2 as the output expression.
Outlines
📚 Introduction to Function Notation with Equations
This paragraph introduces the concept of function notation using equations, as opposed to visual representations like mappings or graphs. It explains the basic structure of function notation, where a function name (e.g., F, G, H) is followed by an input variable (X) and an equals sign, with the output (Y) being calculated based on the input. The paragraph provides an example function, f(X) = (2X^2 - 3) / (X - 4), and emphasizes the importance of understanding the variable part of the equation and the function name as constant. It illustrates how to use the formula to calculate outputs for any given input, highlighting the process of substituting the input value into the formula and performing the calculation.
🔍 Calculating Outputs with Function Notation
This paragraph delves into the process of calculating specific outputs for a given function using the example of f(2). It explains the importance of understanding the order of operations when substituting values into a function. The paragraph demonstrates the calculation step by step, squaring the input, multiplying by the coefficient, and then performing the necessary subtraction and division to arrive at the output, which in this case is -5/2 or -2.5. The explanation serves to clarify how to handle algebraic calculations within function notation and emphasizes common mistakes to avoid.
📘 Applying Function Notation to Various Examples
The paragraph presents several functions, f(X), g(X), and h(X), with different algebraic expressions, including polynomials and absolute values. It then provides examples of calculating outputs for these functions with various inputs, such as f(-5), g(-6), and h(4.1). The calculations are shown in detail, with special attention given to the handling of absolute values and the process of substitution. The paragraph also touches on the concept of functions where the output is independent of the input, as demonstrated with h(X), which always outputs -17 regardless of the input value.
📘 Further Examples of Function Application
Continuing from the previous paragraph, this section provides additional examples of applying function notation with variable inputs. It demonstrates how to replace the input variable X with an expression, such as X + 1, in the function definition and then carry out the calculation. The process involves squaring the expression, distributing coefficients, and combining like terms to arrive at an output that may involve the variable X. This paragraph reinforces the understanding of function notation by showing how to handle more complex inputs and emphasizes the importance of algebraic manipulation within function definitions.
Mindmap
Keywords
💡Function Notation
💡Equation
💡Input
💡Output
💡Variable
💡Order of Operations
💡Absolute Value
💡Polynomial
💡Function Machine
💡Simplifying Expressions
Highlights
Introduction to function notation with equations, emphasizing the importance of understanding the function as a machine with inputs and outputs.
Explanation of the generic formula to calculate outputs for any input in function notation.
Clarification on the role of the variable 'X' in function notation, representing any input value.
Demonstration of calculating the output of a function using a specific example, f(2).
Emphasis on the order of operations and the importance of parentheses in function notation.
Result of the example calculation, showing f(2) equals -5/2.
Introduction of additional functions f, g, and h with different algebraic expressions.
Explanation of absolute value notation and its application in the function g.
Example calculation of f(-5), illustrating the process of substituting the input into the function.
Result of f(-5) calculation, showing the output as 86.
Example calculation of g(-6), including the handling of absolute values.
Result of g(-6) calculation, with the output being -42.
Explanation of the constant function h, which always outputs -17 regardless of the input.
Example of applying function f with a variable input (X + 1), showing the process of substitution.
Result of the variable input example, where f(X + 1) results in the expression 3x^2 + 4x + 2.
Encouragement for viewers to practice applying functions with variable inputs and expressions.
Transcripts
okay so so far we've looked at function
notation but we've done it very visually
with mappings or sets of ordered pairs
or graphs now we're going to look at
maybe the most common application of
function notation and that's with an
equation okay so let's let's recall
first of all function notation you you
write your function name maybe it's F or
G or H whatever F of your input X and
that equals y so f of your input equals
your output okay
F of your input equals your output so
let's let's assume that we have some
function defined using this equation
okay well we'll do some analysis of this
so let's assume that f of X equals a
fraction two x squared minus three over
X minus four okay now this is a formula
to define this function it's not not a
visual representation to tell you that
well this input has this output it tells
you a generic formula to calculate
outputs for any input okay so so what
this is saying is that the way you use a
formula is the you change the value of
the variable to be whatever you want and
then that variable takes that value
everywhere so our variable here is our
input X okay and so the key is anyway
there's an egg's that's the variable
part of this equation that's the
variable f is the function name that
doesn't change right that's not a
variable that's the name of this machine
for which we can put inputs in and
receive outputs so whatever this X is
both of those X's are the same okay
those are both the same so so you can
kind of kind of think of this as think
of it as
don't get hung up on X don't get hung up
on X we use X because that typically
represents input in algebra but just
think of it as think of it as a machine
f with this chute right this is lit just
like a hole to the machine and you can
put whatever you want into that hole
right whatever input you want whatever
input you want put into this chute which
is going to feed it to the F machine and
it's going to output remember F of this
input whatever that happens to be that
whole thing is what your output is and
so the output of this is going to be two
times the input whatever it is squared
minus three over the input minus four
okay don't don't get hung up on X okay
it just means input and it can change
it's a variable so whatever whatever you
feed into the F chute it's going to come
over here into both of these places and
form a calculation to give you a final
final number output now let's do a
particular example with this function
where we actually calculate an output so
so let's do an example let's let's see
if we can determine the value of F of
two and so what this means is F of 2
means Y value means output so what we
need to do is since two is in
parentheses we know that's the input we
need to input 2 to this function by the
way whenever you see f of X equals like
this whenever you see that that is the
definition of the function so in this
case it's the definition of F it's very
important to understand that this
defines your function just like a
picture of a mapping defines what the
function does to an input same thing
here this is how you calculate outputs
for the function f so f of 2 means plug
2 in for X right and that means I have
to replace all of my exes with two
that's my input or 2 goes into the chute
as the input to the F machine
and then it goes in these spots in my
output now it's important to know your
order of operations when you do this so
for example when I say two x-squared you
have to understand what that means it
means you square X whatever it happens
to be whatever number you're plugging in
you square that first and then you
multiply it by two that's why the
parentheses are extremely important here
when you plug in an input if you're not
sure about the order of operations put
parentheses around your input every time
and that should help okay so if we do
that here it looks like two times two
squared minus three over two minus four
and then using the order of operations I
already mentioned if I square 2 I get 4
4 times 2 is 8 8 minus 3 is 5
so my numerator is 5 and I'm bottom 2
minus 4 is negative 2 so 5 divided by
negative 2 is negative it's a negative
fraction negative five-halves so that's
typically how we write that you could
write negative 2.5 often fractions are
preferred improper fractions not mixed
numbers now what does this mean this
means that if we take the input of 2 and
we input it right 2 is the input to the
F function or the F machine think of
this as a machine that has inner
workings if we input 2 to that and 2
goes through these inner workings all of
this happens and the machine finally
outputs negative five-halves ok so
that's kind of a visual way to think of
this and that's how you calculate
outputs for any input with this
particular function now let's practice
some more function notation with
equations and we'll try to throw in some
examples of commonly missed algebraic
calculations
and that's kind of what's tricky about
making a video like this is I can't
cover all the possible calculations you
you should know how to do so so I'm
gonna try to take on the ones that a lot
of people have trouble with or miss
frequently so here are some functions f
of X let's say is 3x squared minus 2x
plus 1 it's a polynomial it's a
trinomial actually because it has three
terms lots of different things we could
say about this G of X is negative 2
times a vertical bar and inside the
vertical bars I have let's say 3x minus
5 so I've got 3x minus 5 inside these
vertical bars we'll talk about those in
a minute plus 4 so do you remember what
the vertical bars mean in mathematics in
this particular case it means what's
called absolute value so we'll see how
to handle that in some cases if you
don't remember absolute value you might
want to go do some reading up on it
alright last let's say we have H of x
equals negative 17 such a simple
function but because it's so simple it
can be confusing
let's do examples now let's consider
let's consider f of negative 5 so that
means negative 5 is input to the F
function which when it's the equation
that means well here's X that means
these X's have to match that X so I need
to replace these X's with negative
negative 5 so we've got 3 times negative
5 quantity squared minus 2 times
negative 5 plus 1 well this is 3 times
25 that's 75 minus 2 times negative 5 is
negative 10 so I've got minus negative
10 which of course is plus positive 10
and then plus 1 so guess 75 plus 10 plus
1 which is 86 so if we input negative 5
to the F
function it outputs 86 if X is negative
5y is 86 for the F function okay so
that's one example with the F function
let's do another let's look at G of
negative 6 so now I go to my G equation
and I substitute negative 6 in for my
input so this X also turns into negative
6 it's about negative 2 times the
absolute value of 3 times negative 6
minus 5 close the absolute value plus 4
so we have absolute value we treat it
kind of like parentheses we figure out
everything inside first so 3 times
negative 6 is negative 18 negative 18
minus 5 is negative 23 keep your
absolute values we haven't dealt with
those yet and keep everything else as
well so now the next thing I need to do
is I need to multiply a negative 2 by
whatever this is so I need to know what
the absolute value of negative 23 is and
the absolute value of negative 23 is
positive 23 again you use your order of
operations so now this turns into
negative 2 times so my absolute value
bars turn into parenthesis negative 2
times positive 23 plus 4 so now we've
got negative 46 plus 4 is negative 42 so
G of negative 6 equals negative 42 if we
input negative 6 to the G machine the
output is negative 42 all right let's do
another one let's go to the H function
let's look at H of four point one that
means my input or my x-value is four
point one so any place there's an X I
replaced with four point one here the
output does not involve X at all it's
completely independent of your choice of
X so no matter what X is
the output here is negative 17 and it's
that simple
okay so H of 4.1 equals negative 17 and
no matter what you plug into this
function your output is negative 17
because X is not involved in the
calculation at all all right let's do
three more quick examples applying each
of these functions with a variable input
okay let's look at let's look at F of
now let's say the input is X plus 1
don't don't be scared by this all this
means is the input is X plus 1 meaning I
look at my definitions Y go to my F
definition which is right here and I
replace all of the X's with X plus 1 so
this turns into 3 times the quantity X
plus 1 squared minus 2 times the
quantity X plus 1 right I'm replacing
both of those with X plus 1 plus 1 and
then I just simplify this now I'm not
gonna get a number as my answer my
answer is going to involve X because my
input involved X as well all right so
the first thing I need to do is square X
plus 1 which that means X plus 1 times X
plus 1 if you need to write that out X
plus 1 squared means X plus 1 times X
plus 1 and you should be able to
multiply that pretty easily but when you
do you get x squared plus 2x plus 1 you
don't remember how to do that you may
need to do some research on X plus 1
times X plus 1 you just simply
distribute all of this out ok then here
I've got minus 2 times X plus 1 so I
need to distribute my negative 2
and then bring down my plus-one all
right and then we're in the end here we
have one more step well distribute our
three and I'm going to go ahead and
combine like terms so I've got three x
squared
that's my only x squared term then I've
got 6 X but then here I've got -2 X so
that's plus 4 X and then last I got 3
but then I've got minus 2 which is so 3
minus 2 is 1 1 plus 1 is 2 and I can't
simplify it any further I've gotten rid
of all my parentheses and combined all
the like terms and so if I input X plus
1 to this function f the output is 3 x
squared plus 4 X plus 2 okay we could do
the same sort of thing with the other
two functions in fact I think that's
good enough you should be able to input
numbers to these functions or
expressions right with variables
themselves and it's no big deal it's the
same process just plug in whatever this
is plug it in as your input which means
replace all the X's with that expression
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