$$x^2 - 4$$ \geq 0 \implies x \leq -2 \text{ or } x \geq 2$$

Question 1

Given $$f(x) = x^2$ + 3x - 10$$ and $$g(x) = x - 2$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(x - 2)^2$ + 3(x - 2) - 10$$

Question 2

If $$f(x) = x^2$ + 4$$ and $$g(x) = 2x - 1$$, what is $$(g \circ f)(x)$$?

Accepted Answer

$$2(x^2 + 4) - 1$

Question 3

Given $$f(x) = \sqrt{x + 1}$$ and $$g(x) = x^2$ - 4$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x^2 - 4 + 1$$ \geq 0 \implies $$x^2$$ \geq 3 \implies x \leq -\sqrt{3} 	ext{ or } x \geq \sqrt{3}$$

Question 4

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is (f$$ \circ g)(x)$$ and its domain?

Accepted Answer

$$(\sqrt{x})^2$ = x$$, \ x \geq 0$

Question 5

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

Accepted Answer

$$f(g(2)) = f(3) = 3^2$ + 2 \cdot 3 = 9 + 6 = 15$$

Question 6

If $$f(x) = x^2$ and g(x) = 2x + 1$, which of the following is (g$$ \circ f)(x)$$?

Accepted Answer

$$2x^2 + 1$

Question 7

Given $$f(x) = x^2$ + 1$$ and $$g(x) = \sqrt{x}$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x \geq 0$$

Question 8

If $$h(x) = \ln(x^2$ + 1)$$, decompose $$h(x) as$$ $$f(g(x))$$ for suitable $$f$$ and $$g.$$

Accepted Answer

$$f(x) = \ln(x), \ g(x) = x^2$ + 1$$

Question 9

Given $$f(x) = x^2$ + 2$$ and $$g(x) = \sqrt{x}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(\sqrt{x})^2$ + 2 = x + 2$$

Question 10

If $$f(x) = x^2$ and g(x) = x + 1$, which of the following is (f$$ \circ g)(x)$$?

Accepted Answer

$$(x + 1)^2$

Question 11

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

Accepted Answer

$$g(f(2)) = g(5) = 5 - 3 = 2$$

Question 12

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

Accepted Answer

$$f(g(-1)) = f(1) = 1^2$ = 1$$

Question 13

Given $$f(x) = x^2$ + 4$$ and $$g(x) = 2x - 1$$, what is $$(f \circ g)(3)$$?

Accepted Answer

$$f(g(3)) = f(5) = 5^2$ + 4 = 25 + 4 = 29$$

Question 14

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is (g$$ \circ f)(x)$$ and its domain?

Accepted Answer

$$g(f(x)) = \sqrt{x^2$}$$, \ x \in \mathbb{R}$

Question 15

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

Accepted Answer

$$x \in \mathbb{R}$$

Question 16

If $$f(x) = \sqrt{x}$$ and $$g(x) = x^2$ - 4$$, what is the domain of $$(f \circ g)(x)$$?

Accepted Answer

$$x^2 - 4$$ \geq 0 \implies x \leq -2 	ext{ or } x \geq 2$$

Function Composition in Precalculus: Concepts, Evaluation, Domain, and Decomposition

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