Q: Given $$f(x) = x - 5$$ and $$g(x) = x^2$ - 1$$, find $$(f + g)(x)$$ and its domain.

$$(f + g)(x) = x^2$ + x - 6$$, \\ \text{Domain: All real numbers}$

Q: Given $$f(x) = 2x + 3$$ and $$g(x) = \frac{x}{x - 4}$$, find $$(f - g)(x)$$ and its domain.

$$(f - g)(x) = 2x + 3 - \frac{x}{x - 4}$$, \\ \text{Domain: All real numbers except } x = 4$

Q: Given $$f(x) = 1 - x$$ and $$g(x) = x + 5$$, find $$(f \circ g)(x)$$ and its domain.

$$(f \circ g)(x) = 1 - (x + 5) = -x - 4$$, \\ \text{Domain: All real numbers}$

Q: Given $$f(x) = 4x + 2$$ and $$g(x) = \frac{1}{x}$$, find $$(f \circ g)(x)$$ and its domain.

$$(f \circ g)(x) = 4\left(\frac{1}{x}\right) + 2 = \frac{4}{x} + 2$$, \\ \text{Domain: All real numbers except } x = 0$

Question 1

Solve the quadratic equation $$x^2 - 6x + 8 = 0$ by factoring.

Accepted Answer

$$x = 2 or$$ $$x = 4$$

Question 2

Solve the quadratic equation $$2x^2 + x = 1$ by factoring.

Accepted Answer

$$x = 1 or$$ $$x = -\frac{1}{2}$$

Question 3

Find the domain of the function $$g(x) = \frac{5x}{x^2$ - 49}.$$

Accepted Answer

All real numbers except $$x = 7$$ and $$x = -7$$

Question 4

Given $$f(x) = x - 5$$ and $$g(x) = x^2$ - 1$$, find $$(f + g)(x)$$ and its domain.

Accepted Answer

$$(f + g)(x) = x^2$ + x - 6$$, \ 	ext{Domain: All real numbers}$

Question 5

Given $$f(x) = 2x + 3$$ and $$g(x) = \frac{x}{x - 4}$$, find $$(f - g)(x)$$ and its domain.

Accepted Answer

$$(f - g)(x) = 2x + 3 - \frac{x}{x - 4}$$, \ 	ext{Domain: All real numbers except } x = 4$

Question 6

The number of births in the U.S. in thousands, $$x$$ years after 2000, is modeled by $$B(x) = -2.6x^2$ + 33x + 4043. $$The number of deaths is $$D(x) = 2x^2$ - 13x + 2428. $$Write a function for the change in population each year.

Accepted Answer

$$C(x) = B(x) - D(x) = -4.6x^2$ + 46x + 1615$$

Question 7

Using the population change function $$C(x) = -4.6x^2$ + 46x + 1615$$, estimate the change in U.S. population in 2014.

Accepted Answer

$$C(14) = -4.6(14)^2$ + 46(14) + 1615 = 1029$$

Question 8

Given $$f(x) = 2x - 1$$ and $$g(x) = x^2$ + 3$$, evaluate $$(f \circ g)(4). $$

Accepted Answer

$$(f \circ g)(4) = 2(4^2$ + 3) - 1 = 33$$

Question 9

Given $$f(x) = 1 - x$$ and $$g(x) = x + 5$$, find $$(f \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = 1 - (x + 5) = -x - 4$$, \ 	ext{Domain: All real numbers}$

Question 10

Decompose $$h(x) = x^2$ + 5 as a $$composition of two functions $$f$$ and $$g$$ such that $$h(x) = f(g(x)).$$

Accepted Answer

$$f(x) = x + 5$$, $$g(x) = x^2$

Question 11

Which of the following is the correct domain for the function $$p(x) = \sqrt{26 - x}$$?

Accepted Answer

$$x \leq 26$$

Question 12

Given $$f(x) = 4x + 2$$ and $$g(x) = \frac{1}{x}$$, find $$(f \circ g)(x)$$ and its domain.

Accepted Answer

$$(f \circ g)(x) = 4\left(\frac{1}{x}ight) + 2 = \frac{4}{x} + 2$$, \ 	ext{Domain: All real numbers except } x = 0$

Question 13

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

Accepted Answer

All real numbers

Question 14

Given $$h(x) = (3x^2$ - 4x + 1)^5$, decompose h(x$$)$$ as f(g(x$$))$$.

Accepted Answer

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

Question 15

Which of the following best describes the domain of the function $$g(x) = x - 3$$?

Accepted Answer

All real numbers

Question 16

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

Accepted Answer

$$(g \circ f)(-1) = 2((-1)^2$ + 3) - 1 = 7$$

Combinations and Compositions of Functions: Precalculus Study Notes

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