Operations on Functions

Evaluating Functions

Given the functions

f (x) = 3 x - 7

and

g (x) = 2 x^{2} + 1

, find the value of

f (g (2))

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Arithmetic of Functions

Suppose that the functions

f

and

g

are defined as follows.

f (x) = \frac{1}{x + 5} g (x) = \frac{8}{x}

Find

\frac{f}{g}

. Then, give its domain using an interval or union of intervals. Simplify your answers.

(\frac{f}{g}) (x) =

Domain of

\frac{f}{g}

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Domain of Composite Functions

If we have two functions,

f (x) = \sqrt{x}

and

g (x) = 2 x + 3

, what is the domain of the composite function

f (g (x))

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Finding the Sum

Given two functions:

f (x) = 3 x^{2} + 2 x + 1

and

g (x) = 2 x^{2} - x + 3

, what is

f (x) + g (x)

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Finding the Difference

Given the functions

f (x) = 2 x - 3

and

g (x) = x^{2} - 5

, find

(f - g) (x)

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Finding the Product

Given the functions

f (x) = 4 x + 3

and

g (x) = 2 x^{2} - 5

, find the product

(f \cdot g) (x)

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Finding the Quotient

Given the functions

f (x) = 3 x^{2} - 2 x + 1

and

g (x) = x - 1

, find the quotient

\frac{f (x)}{g (x)}

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Finding the Domain of the Sum of the Functions

Given the functions

f (x) = \sqrt{x + 2}

and

g (x) = \frac{1}{x - 3}

, find the domain of the sum of the functions

f (x) + g (x)

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Finding the Domain of the Difference of the Functions

Find the domain of the difference of the functions

f (x) = \sqrt{x - 1}

and

g (x) = \frac{1}{x + 2}

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Finding the Domain of the Product of the Functions

Given the functions

f (x) = \frac{1}{x + 2}

and

g (x) = x^{2} - 4

, find the domain of the product of the functions

h (x) = f (x) g (x)

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Finding the Domain of the Quotient of the Functions

Given two functions

f (x) = x^{2} - 4

and

g (x) = 2 x - 4

, find the domain of the quotient function

h (x) = \frac{f (x)}{g (x)}

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Finding Roots (Zeros)

Given two functions

f (x) = x^{2} - 3 x + 2

and

g (x) = - x^{2} + 5 x - 6

. Find the roots of the function

h (x) = f (x) - g (x)

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Identifying Zeros and Their Multiplicities

Given the function f(x) = (x - 2)^2 (x + 3)^3, identify the zeros of the function and their multiplicities.

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Finding the Inverse

Given the function

f (x) = 2 x + 3

, find the inverse function

f^{- 1} (x)

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Finding Maximum Number of Real Roots

Given a polynomial function of degree 5,

f (x) = 2 x^{5} - 3 x^{4} + 2 x^{3} - x^{2} + 3 x - 2

, find the maximum number of real roots that this function can have.

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Function Composition

For

f (x) = \frac{4}{x + 4}

and

g (x) = \frac{3}{x}

, find a.

(f \circ g) (x)

; b. the domain of

f \circ g

(f \circ g) (x) = \frac{4 x}{3 + 4 x}

(Simplify your answer.) b. What is the domain of

f \circ g

? The domain is (Simplify your answer. Type your answer in interval notation. the expression.)

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Finding the Domain and Range

Consider the function

f (x) = \sqrt{x}

and

g (x) = x^{2}

. Find the domain and range of

(f + g) (x)

(f - g) (x)

(f g) (x)

, and

(f / g) (x)

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Evaluating Functions

Arithmetic of Functions

Domain of Composite Functions

Finding the Sum

Finding the Difference

Finding the Product

Finding the Quotient

Finding the Domain of the Sum of the Functions

Finding the Domain of the Difference of the Functions

Finding the Domain of the Product of the Functions

Finding the Domain of the Quotient of the Functions

Finding Roots (Zeros)

Identifying Zeros and Their Multiplicities

Finding the Inverse

Finding Maximum Number of Real Roots

Function Composition

Finding the Domain and Range

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