Q: Given $$f(x) = \frac{x}{x^2$-1}$$, a student claims the domain is $$x \neq 2. Is $$this correct? Why or why not?

No, the correct domain is $$x \neq 1$$ and $$x \neq -1$$ because the denominator is zero at these points.

Question 1

Given the graph of $$G(x) ($$see image 1), what is the domain of $$G(x)$$?

Accepted Answer

$$[-5, 5]$$

Question 2

Given the graph of $$h(z) ($$see image 2), what is the range of $$h(z)$$?

Accepted Answer

$$[-5, 5]$$

Question 3

For the function $$f(t) ($$see image 3), which of the following best describes the domain?

Accepted Answer

$$[-5, 5]$$

Question 4

For the function $$R(p) ($$see image 4), the open circle at $$p=1$$ means:

Accepted Answer

$$R(1) is $$not defined$$

Question 5

Consider the function $$f(x) = \frac{1}{x-5}. $$Which value is not allowed in the domain, and why?

Accepted Answer

$$x=5$$, because the denominator becomes zero

Question 6

What is the domain of $$g(x) = \sqrt{x-3}$$?

Accepted Answer

$$[3, \infty)$$

Question 7

Find the domain of $$k(x) = \frac{1}{x^2$-9}.$$

Accepted Answer

$$(-\infty, -3) \cup (-3, 3) \cup (3, \infty)$$

Question 8

For the function $$m(x) = \sqrt{10-2x}$$, what is the domain?

Accepted Answer

$$(-\infty, 5]$$

Question 9

A pizza shop can make at most 200 pizzas per day. Each pizza sells for $$14.\ What is the practical domain for the revenue function R(x) = 14x$?

Accepted Answer

$$[0, 200]$$

Question 10

A candle is 20 cm tall and burns at 2 cm per hour until it burns out. What is the practical range for the height function $$h(t) = 20 - 2t$$?

Accepted Answer

$$[0, 20]$$

Question 11

Given $$f(x) = \frac{x}{x^2$-1}$$, a student claims the domain is $$x 
eq 2. Is $$this correct? Why or why not?

Accepted Answer

No, the correct domain is $$x 
eq 1$$ and $$x 
eq -1$$ because the denominator is zero at these points.

Question 12

A window washing company charges $$1$$ per window plus a base fee of $$20. $$They can wash a maximum of $$200$$ windows per appointment. What is the practical range for the total cost function $$C(t) = t + 20$$?

Accepted Answer

$$[20, 220]$$

Question 13

Gas costs $$4$$ per gallon and your car has a $$15-$$gallon tank. What is the domain of the total cost function $$C(T) = 4T$$?

Accepted Answer

$$[0, 15]$$

Question 14

A function $$f$$ has domain $$[0, 10]$$ and range $$[2, 8]. Is$$ $$f(12)$$ defined?

Accepted Answer

No, because $$12 is $$not in the domain.

Question 15

A student claims: 'If the domain of a function is all real numbers, then the range must also be all real numbers.' Which of the following disproves this claim?

Accepted Answer

$$f(x) = x^2$ has domain $$(-\infty, \infty)$$ but range $$[0, \infty)$$

Question 16

Given the graph of $$y = \frac{6}{x^2$ - 3x + 3} ($$see image 6), what is the domain?

Accepted Answer

$$(-\infty, \infty)$$

Question 17

For the function $$f(x) = x^2$ + 7$$, what is the domain?

Accepted Answer

$$(-\infty, \infty)$$

Question 18

You bike at $$12$$ mph and can ride at most $$10$$ hours in a day. What is the practical range for the distance function $$d(t) = 12t$$?

Accepted Answer

$$[0, 120]$$

Question 19

For the function $$f(x) = \frac{x}{\sqrt{x-2}}$$, what is the domain?

Accepted Answer

$$(2, \infty)$$

Domain and Range from Graphs – College Algebra Study Guidance

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