$$\{x \mid x \neq 2, x \neq 3\}$$

$$\{x \mid x \neq -3, x \neq -\frac{2}{3}\}$$

$$\{x \mid x \neq 1, x \neq 2\}$$

Question 1

Given $$f(x) = 5x - 1$$ and $$g(x) = 3x!$$, what is $$(f \circ g)(1)$$?

Accepted Answer

$$14$$

Question 2

If $$f(x) = x^2$ + 2x - 5$$ and $$g(x) = 3x + 1$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$9x^2 + 12x - 2$

Question 3

Which of the following is the correct domain for $$(f \circ g)(x) if$$ $$f(x) = \frac{2}{x+4}$$ and $$g(x) = \frac{5}{x-3}$$?

Accepted Answer

$$\{x \mid x 
eq 2, x 
eq 3\}$$

Question 4

Suppose $$f(x) = 3x - 5. $$What is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 5}{3}$$

Question 5

Which of the following functions is one-to-one?

Accepted Answer

$$g(x) = x^5$

Question 6

If $$f(x) = \frac{1}{x+3}$$, what is the domain of $$f \circ f$$?

Accepted Answer

$$\{x \mid x 
eq -3, x 
eq -\frac{2}{3}\}$$

Question 7

Which statement best describes the horizontal-line test?

Accepted Answer

A function is one-to-one if every horizontal line intersects its graph at most once.

Question 8

Given $$f(x) = 2x - 5$$ and $$g(x) = \frac{x + 5}{2}$$, show that $$(f \circ g)(x) = x. $$What does this imply?

Accepted Answer

That $$g is $$the inverse of $$f.$$

Question 9

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, which composition is defined for all real numbers?

Accepted Answer

$$(f \circ g)(x)$$ for $$x \geq 0$$

Question 10

Which of the following is the correct procedure to find the inverse of a one-to-one function $$f(x)$$?

Accepted Answer

Interchange $$x$$ and $$y in$$ $$y = f(x)$$, solve for $$y$$, and check $$f^{-1}(f(x)) = x$ and f(f$$^{-1}$$(x)) = x.$$

Question 11

If $$f(x) = \frac{2}{x+4}$$ and $$g(x) = \frac{5}{x-3}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$\frac{2}{\frac{5}{x-3} + 4}$$

Question 12

Which of the following is true about the graph of a function and its inverse?

Accepted Answer

They are symmetric with respect to the line $$y = x.$$

Question 13

Given the function $$f(x) = \frac{1}{x+3}$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

Question 14

If $$f(x) = x^2$ + 2x - 5$$ and $$g(x) = 3x + 1$$, what is $$(g \circ f)(x)$$?

Accepted Answer

$$3x^2 + 6x - 14$

Question 15

Which of the following is NOT a one-to-one function?

Accepted Answer

$$f(x) = x^2$

Question 16

If $$f(x) = 6\sqrt{x}$$ and $$g(x) = 2x$$, what is $$(f \circ g)(4)$$?

Accepted Answer

$$12\sqrt{2}$$

Question 17

If $$f(x) = \frac{1}{x+3}$$, what value must be excluded from the domain of $$f \circ f$$?

Accepted Answer

$$x = -\frac{2}{3}$$

Question 18

If $$f(x) = x^2$ + 2x - 5$$ and $$g(x) = 3x + 1$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

All real numbers

Question 19

If $$f(x) = \frac{1}{x+3}$$ and $$g(x) = \frac{3}{x-2}$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$\{x \mid x 
eq 1, x 
eq 2\}$$

Composite and Inverse Functions: Precalculus Study Notes

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