$$(f \circ g)(x) = \dfrac{2x}{3 - x}$$, \quad x \neq 0, x \neq 3$

$$x \in \mathbb{R}, x \neq 1, x \neq -1$$

Question 1

Given $$f(x) = \sqrt{3x + 12}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$[-4, \infty)$$

Question 2

Let $$f(x) = x^2$ - 7x$$ and $$g(x) = \dfrac{3x+2}{x^2$-2x-3}. $$What values must be excluded from the domain of $$g(x)$$?

Accepted Answer

$$x = -3$$ and $$x = 1$$

Question 3

If $$f(x) = x-5$$ and $$g(x) = x^2$-1$$, what is $$(f+g)(x)$$?

Accepted Answer

$$x^2 - x - 6$

Question 4

Given $$f(x) = 2x+1$$ and $$g(x) = 3x-2$$, find $$(f \circ g)(x).$$

Accepted Answer

$$6x - 1$$

Question 5

Let $$f(x) = \dfrac{2}{x-1}$$ and $$g(x) = \dfrac{3}{x}. $$What is $$(f \circ g)(x)$$ and its domain?

Accepted Answer

$$(f \circ g)(x) = \dfrac{2x}{3 - x}$$, \quad x 
eq 0, x 
eq 3$

Question 6

Given $$f(x) = \sqrt{x-1}$$ and $$g(x) = x+3$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \geq -2$$

Question 7

If $$f(x) = x-5$$ and $$g(x) = x^2$-1$$, what is the domain of $$\left(\dfrac{f}{g}\right)(x)$$?

Accepted Answer

$$x \in \mathbb{R}, x 
eq 1, x 
eq -1$$

Question 8

Let $$f(x) = 2x-1$$ and $$g(x) = x^2$ + x - 2. $$What is $$(f \cdot g)(x)$$?

Accepted Answer

$$2x^3$$ + $$x^2 - 5x + 2$

Question 9

Which of the following best describes the domain of the composite function $$(f \circ g)(x)$$?

Accepted Answer

All $$x in $$the domain of $$g$$ such that $$g(x) is in $$the domain of $$f$$

Question 10

Decompose $$h(x) = (3x^2$ - 4x + 1)^5$ as a composition of two functions f$ and g$.

Accepted Answer

$$f(x) = x^5$, \quad g(x) = 3x^2$ - 4x + 1$$

Question 11

Given $$f(x) = \dfrac{18}{x}$$ and $$g(x) = \dfrac{18}{x}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$\dfrac{18}{\dfrac{18}{x}}$$

Question 12

Suppose $$f(x) = x^2$ - 1$$ and $$g(x) = x - 5. $$What is $$(g \circ f)(2)$$?

Accepted Answer

$$-2$$

Question 13

Which of the following is NOT a valid reason to exclude a value from a function's domain?

Accepted Answer

The value makes the function output a very large number

Question 14

Let $$f(x) = \sqrt{x^2$ + 1}$$ and $$g(x) = x^3$. What is the domain of (f$$ \circ g)(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 15

Given $$f(x) = x^2$ - 1$$ and $$g(x) = x - 5$$, what is $$(f \cdot g)(x)$$?

Accepted Answer

$$(x^2 - 1)(x - 5$$)$$

Combinations and Composition of Functions: Domains and Operations

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