Question 1

Which of the following is the equation for the identity function, and what is its main property?

Accepted Answer

$$f(x) = x$$, and it leaves every real number unchanged.

Question 2

The graph of the squaring function $$f(x) = x^2$ is called a parabola. Which real-world application relies on the reflection property of parabolas?

Accepted Answer

Satellite dishes

Question 3

Which of the following functions has a domain of all real numbers except $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 4

The cubing function $$f(x) = x^3$ has a special point at the origin. What is this point called, and why is it significant?

Accepted Answer

Point of inflection; the graph changes curvature at this point.

Question 5

Which function is even, and what does this imply about its graph?

Accepted Answer

$$f(x) = |x|$$; its graph is symmetric with respect to the y-axis.

Question 6

A medical imaging device uses the exponential function $$f(x) = e^x to $$model radioactive decay. What is a key property of the exponential function relevant to this application?

Accepted Answer

It grows faster than any polynomial function as $$x$$ increases.

Question 7

Which of the following statements about the natural logarithm function $$f(x) = \ln(x) is $$true?

Accepted Answer

It increases very slowly as $$x$$ increases.

Question 8

A sound engineer analyzes a signal using the sine function $$f(x) = \sin(x). $$What is a unique property of the sine function that is useful in this context?

Accepted Answer

It is periodic with period $$2\pi.$$

Question 9

Which function has a 'corner' at the origin, and what does this mean for its graph?

Accepted Answer

$$f(x) = |x|$$; the graph changes direction abruptly at the origin.

Question 10

The greatest integer function $$f(x) = \lfloor x \rfloor is $$used in a payroll system to round down hours worked to the nearest integer. What is a key property of this function?

Accepted Answer

It has a jump discontinuity at every integer value of $$x.$$

Question 11

Which of the following functions has two horizontal asymptotes, and what are their equations?

Accepted Answer

$$f(x) = \frac{1}{1 + e$$^{-x}$$}$$; $$y = 0$$ and $$y = 1$$

Question 12

A satellite dish is designed using the reciprocal function $$f(x) = \frac{1}{x}. $$What is a significant property of this function that is useful in this context?

Accepted Answer

It has a reflection property similar to parabolas.

Question 13

Suppose $$f(x) = -|x| - 1. $$Which of the following statements is true about its graph?

Accepted Answer

It is the graph of $$|x|$$ reflected over the x-axis and shifted down 1 unit.

Question 14

Which of the following functions is odd, and what does this imply about its graph?

Accepted Answer

$$f(x) = x^3$; its graph is symmetric with respect to the origin.

Question 15

A biologist uses the logistic function $$f(x) = \frac{1}{1 + e$$^{-x}$$} to $$model population growth. What is a key feature of this function?

Accepted Answer

It has two horizontal asymptotes at $$y = 0$$ and $$y = 1.$$

Question 16

Which function is used to model periodic phenomena such as sound waves and tides?

Accepted Answer

$$f(x) = \sin(x)$$

Question 17

If $$f(x) = \sqrt{x}$$, what is the domain of $$f$$?

Accepted Answer

$$[0, \infty)$$

Question 18

Which of the following best describes the behavior of $$f(x) = \ln(x) as$$ $$x$$ approaches 0 from the right?

Accepted Answer

$$f(x)$$ approaches $$-\infty$$

Question 19

A business uses the step function $$f(x) = \lfloor x \rfloor to $$determine shipping costs based on weight. If a package weighs 3.7 kg, what value does the function assign?

Accepted Answer

3

Question 20

Which of the following functions is not continuous on its entire domain?

Accepted Answer

$$f(x) = \lfloor x \rfloor$$

Twelve Basic Functions and Their Properties in Precalculus

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