Operations on Functions
Summary
TLDRThis video script introduces the fundamental operations of functions, including addition, subtraction, multiplication, and division, with the stipulation that division by zero is undefined. It demonstrates these operations step by step using specific functions, such as 'v(x) - p(x)', 'f(x) * p(x)', 'v(x) / g(x)', and 'f(x) + g(x)'. The script also includes examples of factoring and simplifying expressions, concluding with evaluating functions at a specific point, x = 3, to yield numerical results.
Takeaways
- ✏️ The objective is to perform fundamental operations with functions.
- ➕ The sum of two functions, f(x) and g(x), is given by f(x) + g(x).
- ➖ The difference between two functions is given by f(x) - g(x).
- ✖️ The product of two functions is f(x) multiplied by g(x).
- ➗ The quotient of two functions is f(x) divided by g(x), with the condition that g(x) is not equal to zero.
- 🔍 For subtraction, substitute the given functions and combine like terms.
- 🧮 For multiplication, use the distributive property to expand and simplify the expression.
- 🔄 For division, factor the numerator and denominator, then simplify by canceling common factors.
- 🔢 When adding functions, combine like terms after substituting the given values.
- ✍️ For specific values, substitute the given value of x into the functions and simplify.
Q & A
What are the fundamental operations that can be performed on functions f and g?
-The fundamental operations on functions f and g include the sum (f + g), difference (f - g), product (f * g), and quotient (f / g), provided that g(x) is not equal to zero to avoid division by zero.
What is the definition of the sum of two functions, f and g?
-The sum of functions f and g, denoted as f + g, is defined as f(x) + g(x) for all x in the domain of both f and g.
How is the difference between two functions, f and g, represented?
-The difference between functions f and g is represented as f - g, which is equal to f(x) - g(x) for all x in the domain of both f and g.
What is the definition of the product of two functions, f and g?
-The product of functions f and g is defined as f * g, which equals f(x) * g(x) for all x in the domain of both functions.
What is the definition of the quotient of two functions, f and g, and what condition must be met?
-The quotient of functions f and g is defined as f / g, which equals f(x) / g(x) for all x in the domain of both functions, provided that g(x) is not zero to avoid an undefined function.
In the given transcript, what is the expression for v(x) - p(x)?
-The expression for v(x) - p(x) is x^2 - 4x - 5 - (2x - 7), which simplifies to x^2 - 6x + 2 after combining like terms.
How is the product of f(x) and p(x) calculated in the transcript?
-The product f(x) * p(x) is calculated by multiplying (x + 3) with (2x - 7), resulting in 2x^2 - 7x + 6x - 21, which simplifies to 2x^2 - x - 21.
What is the quotient of v(x) divided by g(x) in the transcript, and how is it simplified?
-The quotient v(x) / g(x) simplifies to (x^2 - 4x - 5) / (x^2 - 25), which factors and simplifies to (x - 5)(x + 1) / (x - 5)(x + 5), and further simplifies to (x + 1) / (x + 5) after canceling out the common factor (x - 5).
What is the expression for f(x) + g(x) in the transcript, and how is it simplified?
-The expression for f(x) + g(x) is (x + 3) + (x^2 - 25), which simplifies to x^2 + x - 22 after combining like terms.
How is the value of f(x) + g(x) evaluated when x is substituted with 3 in the transcript?
-When x is substituted with 3, the value of f(x) + g(x) becomes 3^2 + 3 - 22, which simplifies to 9 + 3 - 22, resulting in -10.
In the transcript, what is the process for evaluating f(3) + g(3), and what is the result?
-The process involves substituting 3 for x in both f(x) and g(x), resulting in (3 + 3) + (3^2 - 25), which simplifies to 6 + 9 - 25, and the final result is -10.
Outlines
📚 Fundamental Operations on Functions
This paragraph introduces the basic arithmetic operations that can be performed on functions. It defines the sum, difference, product, and quotient of two functions, f and g, within their common domain. The paragraph also highlights the importance of ensuring the denominator is not zero in the quotient operation to avoid undefined results. It then proceeds with examples of subtraction and multiplication of functions, demonstrating the process of replacing function values and simplifying expressions to find the resulting function.
🔍 Detailed Example of Function Operations
The second paragraph provides a step-by-step walkthrough of performing arithmetic operations on functions, specifically focusing on subtraction, multiplication, and division. It begins with an example of subtracting function p from function v, then moves on to multiplying function f by function p. The paragraph also includes an example of dividing function v by function g, emphasizing the importance of factoring to simplify the division. Each step is clearly explained, from the initial definition of the operation to the final simplification of the expression, resulting in the new function representations.
Mindmap
Keywords
💡Functions
💡Sum of Functions
💡Difference of Functions
💡Product of Functions
💡Quotient of Functions
💡Factoring
💡Combining Like Terms
💡Binomials
💡Substitution
💡Undefined Function
Highlights
Introduction to fundamental operations in functions, including addition, subtraction, multiplication, and division.
Definition of the sum of two functions, f and g, as f(x) + g(x).
Explanation of the difference between two functions, f(x) - g(x).
Description of the product of two functions, f(x) * g(x).
Clarification on the quotient of two functions, f(x) / g(x), with the condition that g(x) ≠ 0.
Example calculation of the difference between functions v(x) - p(x).
Step-by-step solution for v(x) - p(x), including substitution and rearrangement of terms.
Example of multiplying functions f(x) and p(x), with detailed steps for binomial multiplication.
Combining like terms after multiplying functions f(x) and p(x).
Division of functions v(x) by g(x), with factoring and simplification.
Cancellation of common factors in the division of functions v(x) / g(x).
Addition of functions f(x) and g(x), with a focus on rearranging and combining terms.
Substitution of a specific value (3) into the function f(x) + g(x) to find f(3) + g(3).
Verification of the result for f(3) + g(3) through direct substitution into f(x) and g(x).
Final confirmation of the result for f(3) + g(3) being -10 through two different methods.
Transcripts
our objective is to perform the
fundamental operations
in functions let f and g be functions
each function is defined for all x in
the domains of both
f and g letter a the sum of
f and g is f plus g of x equals f of
x plus g of x letter b
difference of f and g is f minus g
of x is equal to f of x minus
g of x letter c product of
f and g is f times g of x
is equal to f of x times g of x
and the quotient of f and g
is f over g of x equals
f of x divided by g of x provided that g
of x is not equal to zero because when g
of x
our denominator is equal to zero the
function will become
undefined let us consider the following
functions
and let us solve number one v minus p
of x our first step is definition of
subtraction
functions so this will become v of x
minus p of x
copy the left hand side of the equation
our next step is
replace v of x and p of x by the given
values
so our v of x is x squared
minus four x minus five minus
our p of x is two x
minus seven make sure you enclose that
in
parenthesis copy this our third step
will be
rearranging the terms so this will
become x
squared minus 4x
minus 2x minus 5
this will become plus 7 combine like
terms
x squared negative 4x minus 2x
negative 6x negative 5 plus 7
positive 2. and this is our answer for
v minus p of x
next one let us try f times
p of x so first step again
definition of this time multiplication
of functions so this will become
f of x times
p of x copy this
next step we'll replace f of
x and p of x by the given values
so our f of x is
x plus 3 times our p
of x is 2x minus
7 copy this third step
multiply the binomials so we have
x times 2x is 2x squared
x times negative seven is negative seven
x three times two x
is positive six x and three
times negative seven is negative twenty
one
let us combine like terms so this will
still become
2x squared negative 7x plus 6x
negative x copy minus
21. this will be our final answer for
f times p of x
okay another example this time v
divided by g of x so our first step
definition of this time division
so this will become v of x
divided by g of x
copy this we replace
v of x and g of x by the given values
so our v of x x squared minus four
x minus five divided by
all over our g of x
x squared minus twenty
copy this then we factor the numerator
and the denominator
so the knowledge in factoring trinomials
as well as binomials
is an advantage in this lesson
the factors of x squared minus 4x minus
5
are x minus 5 and x
plus 1 all over the factors of
x squared minus 25 are x minus five
and x plus five cancel out the common
factors
so x minus five and x minus five
so we will be left with x plus one
all over x plus 5. so this is our answer
for
v divided by g of x
okay next let us try f plus g
of x so our first step definition of
this time
addition functions so we have f
of x plus g of x
copy this then we replace again
our f of x will be replaced with x plus
3
plus our g of x is x
squared minus 25 copy this
and then rearrange the terms this will
be the first one
x squared plus x plus 3
minus 25 and then we combine like terms
x squared plus x 3 minus 25
is negative 22. so this is our answer
for
f plus g of x so take note of this
answer
x squared plus x minus 22 let's have the
next one
so i'm still evaluating here f plus g
but this time
instead of having x here i have three
so our previous answer x squared plus x
minus 22
is actually the same as here so let's
proceed
copy this our next step will be
to substitute 3 for x so since we have
here
3 we're going to substitute 3 in all our
x's here
so this will become 3 squared
plus 3 copy minus 22
and then we simplify 3 squared is 9
9 plus 3 is 12 and 12 minus 22
is negative 10. so that will be our
answer for
f plus g of 3. now there is another
solution here
so we should also arrive with negative
10.
so we'll see first step
so this will be f of 3
plus g of three
so copy this our next step will be to
substitute three
for x in both f of x
and g of x so we're going to substitute
3
in our f of x so substituting 3
here this will become 3 plus 3
then plus now for g
substituting 3 here this will become
3 squared minus 25
copy f plus g of 3 and then we simplify
3 plus 3 is 6 3 squared
is 9 and minus
25 copy this let us simplify further
6 plus 9 15 copy
minus 25 15 minus 25
is negative 10. so we have the same
answer as the previous one this is our
answer for
f plus g of 3.
gets
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