Q: Which piecewise function matches the graph shown below? [Image 1]

$$y = \begin{cases} -x + 3, & -1 \leq x \leq 2 \\ -2x + 3, & 2 \u003c x \leq 3 \end{cases}$$

Question 1

Given the function $$g(x) = -x + 5$$, what is the domain and range of $$g(x)$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$(-\infty, \infty)$$

Question 2

Which piecewise function matches the graph shown below?

[Image 1]

Accepted Answer

$$y = \begin{cases} -x + 3, & -1 \leq x \leq 2 \ -2x + 3, & 2 \u003c x \leq 3 \end{cases}$$

Question 3

For the function $$f(x) = \frac{5x^2$ - 9}{x - 3}$$, what are all possible asymptotes?

Accepted Answer

$$x = 3$$ and $$y = 5x + 15$$

Question 4

Solve for $$x$$: $$\log_3$(x^2 - 4) = 2$.

Accepted Answer

$$x = \pm 5$$

Question 5

Given $$f(x) = -5 + 7x$$, which of the following is its inverse function?

Accepted Answer

$$f^{-1}(x$$) = \frac{x + 5}{7}$$

Question 6

Is the following trigonometric equation an identity?

$$\frac{1 + \sin A}{\cos A} = \frac{\cos A}{1 - \sin A}$$

Accepted Answer

Not an identity

Question 7

Which table correctly shows the values of trigonometric functions for the given angles?

Accepted Answer

[Image 6]

Question 8

Given the graph of $$h(x)$$, which graph represents $$h(x + 3)$$?

Accepted Answer

[Image 10]

Question 9

Which of the following pairs of $$u(x)$$ and $$v(x) is $$correct for $$u(v(x)) = (x^3$ - 5)^4$?

Accepted Answer

$$u(x) = x^4$, v(x$$) = $$x^3 - 5$

Question 10

Simplify the expression $$\left(\frac{10}{5}\$$right)^3$$.

Accepted Answer

8

Question 11

A certain species of fish in a lake doubles its population every 24 hours. The initial population is 200. If the population $$t$$ hours after the initial observation is $$p(t) = 200 \cdot 2$$^{t/24}$$, what is the population 3 days after the initial observation?

Accepted Answer

1600

Question 12

Given $$\log_c a = 0.45$$ and $$\log_c b = 0.65$$, find $$\log_c \left(\frac{a}{b}\right).$$

Accepted Answer

$$-0.20$$

Question 13

Identify the intervals where the function $$h(x) = |x + 4| is $$one-to-one and thus has an inverse.

Accepted Answer

$$(-\infty, -4)$$ and $$(4, \infty)$$

Question 14

Which of the following is equivalent to $$\csc$$^{-1}$$(y)$$ for $$y 
eq 0$$?

Accepted Answer

$$\sin$$^{-1}$$\left(\frac{1}{y}\right)$$

Question 15

Prove the limit statement: $$\lim_{x 	o 2}(7 - x) = 5. $$Which of the following choices for $$\delta (in $$terms of $$\varepsilon) $$correctly proves this limit?

Accepted Answer

$$\delta = \varepsilon$$

Question 16

Evaluate the limit: $$\lim_{x 	o -\infty} 11e^x \cos x.$$

Accepted Answer

0

Question 17

Which of the following graphs represents a function that is continuous on the interval $$(0, 15)$$?

Accepted Answer

[Image 14]

Calculus Midterm Study Guidance – Piecewise Functions

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