Q: Let $$u(x) = \sqrt{2-x}$$ and $$v(x) = \dfrac{1}{x-4}. $$What is the domain of $$(u \circ v)(x)$$?

$$x \neq 4$$ and $$2 - \dfrac{1}{x-4} \geq 0$$

Q: Given $$u(x) = \sqrt{2-x}$$ and $$v(x) = \dfrac{1}{x-4}$$, what is the domain of $$(v \circ u)(x)$$?

$$x \u003c 2$$ and $$\sqrt{2-x} \neq 4$$

Question 1

Which of the following correctly describes the domain of the function $$f(x)$$ defined as the semicircle of radius 2 above the $$x-$$axis centered at $$(1,0)$$?

Accepted Answer

$$x \in [-1,3]$$

Question 2

Given the function $$q(x)$$ defined by $$q(x) = \begin{cases} x+2, & x2 \end{cases}$$ and $$k(x) = x-1$$, what is $$(q \circ k)(x) as a $$piecewise function?

Accepted Answer

$$\begin{cases} x+1, & x3 \end{cases}$$

Question 3

For the function $$s(x) = \dfrac{Ax+B}{x-1} + C$$ with $$x-$$intercept $$x=2$$, $$y-$$intercept $$y=-1$$, and horizontal asymptote $$y=3$$, what is the value of $$C$$?

Accepted Answer

$$C = 3$$

Question 4

Solve the inequality $$|2x-5| \leq 7.$$

Accepted Answer

$$x \in [-1, 6]$$

Question 5

Find the exact range of $$f(x) = -2x^2$ + 8x - 5.$$

Accepted Answer

$$(-\infty, 3]$$

Question 6

Let $$u(x) = \sqrt{2-x}$$ and $$v(x) = \dfrac{1}{x-4}. $$What is the domain of $$(u \circ v)(x)$$?

Accepted Answer

$$x 
eq 4$$ and $$2 - \dfrac{1}{x-4} \geq 0$$

Question 7

Given $$u(x) = \sqrt{2-x}$$ and $$v(x) = \dfrac{1}{x-4}$$, what is the domain of $$(v \circ u)(x)$$?

Accepted Answer

$$x \u003c 2$$ and $$\sqrt{2-x} 
eq 4$$

Question 8

Why do the domains of $$(u \circ v)(x)$$ and $$(v \circ u)(x)$$ differ for $$u(x) = \sqrt{2-x}$$ and $$v(x) = \dfrac{1}{x-4}$$?

Accepted Answer

Because the order of composition changes which function's domain restrictions apply first.

Question 9

Define a piecewise function $$T(x) so $$that $$T(-2) = 3$$, $$T(0) = 0$$, $$T(4) = -2$$, and $$T is $$linear on each interval $$(-\infty, -2]$$, $$[-2, 0]$$, $$[0, 4]$$, $$[4, \infty)$$ with constant values outside $$[-2, 4]. $$What is $$T(x)$$ for $$x \u003c -2$$?

Accepted Answer

$$T(x) = 3$$

Question 10

Given the function $$f(x)$$ defined as the line of slope $$-2$$ through $$(0,1)$$ for $$x \u003c 0$$, what is its explicit formula for $$x \u003c 0$$?

Accepted Answer

$$f(x) = 1 - 2x$$

Question 11

For the function $$s(x) = \dfrac{Ax+B}{x-1} + C$$ with $$x-$$intercept $$x=2$$, $$y-$$intercept $$y=-1$$, and horizontal asymptote $$y=3$$, what is the value of $$A$$?

Accepted Answer

$$A = 3$$

Question 12

For the function $$s(x) = \dfrac{Ax+B}{x-1} + C$$ with $$x-$$intercept $$x=2$$, $$y-$$intercept $$y=-1$$, and horizontal asymptote $$y=3$$, what is the value of $$B$$?

Accepted Answer

$$B = -1$$

Question 13

Solve the inequality $$\left| \dfrac{x+1}{x-2} \right| \u003e 2 ($$excluding points not in the domain). Which interval(s) describe the solution?

Accepted Answer

$$x  4$$

Question 14

The range of $$f(x) = -2x^2$ + 8x - 5 is$$:

Accepted Answer

$$(-\infty, 3]$$

Question 15

For the function $$f(x) = -2x^2$ + 8x - 5$$, what is the vertex of the parabola?

Accepted Answer

$$(2, 3)$$

Question 16

Given $$u(x) = \sqrt{2-x}$$, what is the domain of $$u(x)$$?

Accepted Answer

$$x \leq 2$$

Question 17

Given $$v(x) = \dfrac{1}{x-4}$$, what is the domain of $$v(x)$$?

Accepted Answer

$$x 
eq 4$$

Question 18

For the piecewise function $$T(x)$$ defined so that $$T(-2) = 3$$, $$T(0) = 0$$, $$T(4) = -2$$, and $$T is $$linear on $$[-2,0]$$, what is the formula for $$T(x) on$$ $$[-2,0]$$?

Accepted Answer

$$T(x) = -\dfrac{3}{2}x + 3$$

Question 19

For the function $$f(x)$$ defined as the semicircle of radius 2 above the $$x-$$axis centered at $$(1,0)$$, what is its explicit formula for $$0 \leq x \leq 3$$?

Accepted Answer

$$f(x) = \sqrt{4 - (x-1)^2$}$$

College Algebra: Piecewise Functions, Function Composition, and Related Concepts

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