Function Operations
Summary
TLDRThe video script explains the operations on two functions, f(x) = 2x + 5 and g(x) = x^2 - 4. It demonstrates how to find the sum (f + g = x^2 + 2x + 1), difference (f - g = -x^2 + 2x + 9), and product (f * g = 2x^3 + 5x^2 - 8x - 20) of the functions. It then discusses the domain of these functions, highlighting that polynomials have a domain of all real numbers, while fractions require excluding values that make the denominator zero. Examples are given to calculate specific values of f and g for certain x inputs, emphasizing the process of substitution and arithmetic operations.
Takeaways
- 🔢 The function f(x) = 2x + 5 and g(x) = x^2 - 4 are given, and their sum f + g results in x^2 + 2x + 1 after combining like terms.
- ➖ The difference f - g is calculated as -x^2 + 2x + 9 by subtracting g(x) from f(x) and simplifying.
- 🔗 The product f * g is found by multiplying the two functions using the FOIL method, resulting in 2x^3 + 5x^2 - 8x - 20.
- 🌐 The domain of f + g, f - g, and f * g is all real numbers since they are polynomials without restrictions.
- 🚫 The domain of f / g is restricted because it involves a fraction; x cannot be the values that make the denominator zero (x ≠ -2, 2).
- 📐 The domain for functions involving fractions is determined by setting the denominator equal to zero to find the vertical asymptotes.
- 📝 Interval notation is used to express the domain of functions with restrictions, excluding the values that make the denominator zero.
- 🔢 To find f(a) + g(b), substitute 'a' into f(x) and 'b' into g(x), then add the results.
- 🔄 To calculate f(a) * g(b), first find the values of f(a) and g(b) separately, then multiply them together.
- 📘 The script provides a comprehensive guide on how to perform operations on functions and determine their domains.
Q & A
What is the function f(x) as described in the transcript?
-The function f(x) is described as f(x) = 2x + 5.
What is the function g(x) as described in the transcript?
-The function g(x) is described as g(x) = x^2 - 4.
How do you find the sum of the functions f(x) and g(x)?
-You add the two functions together, which results in (2x + 5) + (x^2 - 4), and then combine like terms to get x^2 + 2x + 1.
What is the result of f(x) - g(x) according to the transcript?
-The result of f(x) - g(x) is -x^2 + 2x + 9.
How is the product of f(x) and g(x) calculated?
-The product f(x) * g(x) is calculated by multiplying (2x + 5) with (x^2 - 4) using the FOIL method, resulting in 2x^3 + 5x^2 - 8x - 20.
What is the domain of the function f(x) + g(x)?
-The domain of f(x) + g(x) is all real numbers, as it is a polynomial without restrictions.
What is the domain of the function f(x) - g(x)?
-The domain of f(x) - g(x) is also all real numbers, as it is a polynomial without restrictions.
What is the domain of the function f(x) * g(x)?
-The domain of f(x) * g(x) is all real numbers, as it is a polynomial without restrictions.
How do you determine the domain of f(x) / g(x)?
-The domain of f(x) / g(x) is all real numbers except where the denominator g(x) equals zero, which is when x = -2 or x = 2.
What is the interval notation for the domain of f(x) / g(x)?
-The interval notation for the domain of f(x) / g(x) is (-∞, -2) U (-2, 2) U (2, ∞).
What is the value of f(2) + g(3) as described in the transcript?
-The value of f(2) + g(3) is calculated as (4*2 + 5) + (8 - 3^2) which equals 13 - 1, resulting in 12.
What is the value of f(-2) * g(2) according to the transcript?
-The value of f(-2) * g(2) is calculated as (4*(-2) + 5) * (8 - 2^2) which equals (-3) * 4, resulting in -12.
Outlines
📘 Operations and Domain of Functions
This paragraph explains the operations on two functions, f(x) = 2x + 5 and g(x) = x^2 - 4. It details how to find the sum (f + g = x^2 + 2x + 1), difference (f - g = -x^2 + 2x + 9), and product (f * g = 2x^3 + 5x^2 - 8x - 20) of the functions. It also discusses the domain of these functions, stating that since they are polynomials without radicals or fractions, their domain is all real numbers (-∞, ∞). However, for the division f/g, where f/g = (2x + 5) / (x^2 - 4), the domain is restricted to all real numbers except x = ±2, as these values make the denominator zero, leading to undefined function values. The domain is expressed in interval notation as (-∞, -2) U (-2, 2) U (2, ∞).
🔢 Evaluating Functions at Specific Points
The second paragraph demonstrates how to evaluate functions at specific points. It uses the functions f(x) = 4x + 5 and g(x) = 8 - x^2 to calculate f(2) + g(3) and f(-2) * g(2). For f(2) + g(3), it substitutes x with 2 in f(x) and x with 3 in g(x), resulting in 13 - 1 = 12. For f(-2) * g(2), it first calculates f(-2) = -3 and g(2) = 4, then multiplies these values to get -12. The process involves substituting the given x values into the function's formulas and performing the arithmetic operations to find the results.
Mindmap
Keywords
💡Function
💡Polynomial
💡Domain
💡Range
💡Vertical Asymptote
💡Factoring
💡Interval Notation
💡Combining Like Terms
💡FOIL Method
💡Substitution
Highlights
Function f(x) is defined as 2x + 5.
Function g(x) is defined as x^2 - 4.
Sum of functions f and g results in x^2 + 2x + 1 after combining like terms.
Difference of functions f and g results in -x^2 + 2x + 9 after distributing and combining constants.
Product of functions f and g is calculated using the FOIL method, resulting in 2x^3 + 5x^2 - 8x - 20.
Domain of f + g, f - g, and f * g is all real numbers since they are polynomials without restrictions.
Domain of f / g is restricted to all real numbers except x = -2 and x = 2, as these make the denominator zero.
Domain of a function with a fraction is determined by setting the denominator equal to zero to find restrictions.
Vertical asymptotes at x = -2 and x = 2 for the function f / g, indicating points of discontinuity.
Domain of f / g is expressed in interval notation as (-∞, -2) U (-2, 2) U (2, ∞).
Function f(x) = 4x + 5 is given as an example for evaluating f(2).
Function g(x) = 8 - x^2 is used to evaluate g(3).
f(2) + g(3) is calculated by substituting x values into their respective functions and summing the results.
f(-2) * g(2) is computed by finding the values of f at x = -2 and g at x = 2, then multiplying these values.
The process of evaluating functions at specific points involves substitution and arithmetic operations.
Transcripts
let's say that f of x
is equal to two x plus five
and g of
x let's say g of x is
x squared minus four
perform the indicated operations
so what is f plus g
what's the sum of the two functions all
you gotta do is add them 2x plus five
plus x squared minus four
and combine like terms
so all we can combine is five and
negative four
which adds up to one so it's x squared
plus two x
plus one
and so that's the sum of f and g
now what about f minus g
f is two x plus five
and then it's gonna be minus
x squared minus four
so this is going to be negative x
squared
plus two x
and then we have five
minus negative four
five minus negative four is like five
plus four
which is nine
so that's equal to f minus g
now what about f times g
this is just going to be two x plus 5
times x squared minus 4.
and we can go ahead and foil it 2x times
x squared
that's 2x cubed
and then 2x times negative 4 that's
negative 8x
and then we have 5 times
x squared that's
5x squared
and then 5 times negative 4 it's
negative 20.
so in standard form it's 2x cubed
plus 5x squared
minus 8x minus 20.
now what is the domain
of the three functions that we found
let's start with f plus g
what is the domain for that
whenever you have a polynomial be it a
binomial trinomial or many terms where
you don't have any fractions no radicals
or
logarithmic functions
the domain will be all real numbers
there's no restrictions on the value of
x in this expression x can be anything
so for these three functions f plus g f
minus g and f times g
we don't have any fractions or radicals
so the domain is all real numbers it's
negative infinity to infinity
now what about f
divided by g
what's the domain for this
f divided by g is simply two x plus five
divided by x squared minus 4.
now that we have a fraction
the domain is restricted
is not all real numbers
to find the domain
you want to find the x values
that do not exist
the values that x cannot be
to do that
set the denominator equal to zero
x squared minus four cannot be zero if
it is we're going to get a vertical
asymptote which means it's undefined at
that point
now we can factor x squared minus four
it's x plus two times x minus two
so therefore x
cannot equal negative two
and it can't equal two these are the
vertical asymptotes
but they're also
infinite discontinuities
there's no we can't plug in negative two
for x the function will be undefined
anytime you have a zero on the bottom
it's undefined
so how can we write the domain using
interval notation if x cannot equal
negative two or two
it's going to be from negative infinity
to negative two
union negative two to two
union
two to infinity
so here's another example let's say
if we have the function one over x minus
three
x minus three the denominator cannot be
zero
so x cannot be three
therefore the domain is everything
except three that's how you write it
let's say if we have this
x
cannot equal 4
and it can't equal
negative 3. just change the sign
if you set x minus four to zero x
you'll get four facts
now to write the domain it's gonna be
negative infinity to negative three
negative three comes before four
union negative three is a four
union 4 to infinity
so that's the domain for this function
so let's say that
f of x
is equal to 4x plus 5.
and g of x
let's say it's equal to 8
minus x squared
what is f of 2
plus g of three
to find f of two
we need to replace x with two in the
equation four x plus five
so that's four times two plus five
and that is a terrible looking two
to find g of 3 we need to plug it into
that equation so it's plus
8 minus
3 squared
and then let's add 4 times 2 is 8
and 3 squared
is nine eight plus five is thirteen
eight minus nine is negative one
thirteen minus one is twelve
so that's the value of f of two plus g
of three
you just find the two values
and you add them
let's try another example
so what is
f of negative 2
multiplied
by
g of two
so let's do it separately let's find f
of negative two first
using this formula
so it's gonna be four times negative two
plus five
which is negative eight plus five
and so that's negative three
now let's calculate g of two
using this equation
so that's eight minus two squared
two squared is four eight minus four is
four
so now that we have these two values
let's replace f of negative two which
is negative three and let's replace g of
two with four
so negative three times four is negative
twelve
so that's the value of f of negative two
times g of negative two i mean times g
of positive two
so all you gotta do is simply
find the values
and then multiply them
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