Why does the function $f(x) = x^2 + 5$ not have an inverse over the set of all real numbers?

The function's graph does not have a y-intercept.

There are at least two different inputs that produce the same output.

The function is not defined for all possible input values.

The output of the function is always positive.

Which statement best explains why the function $g(x) = \cos(x)$ does not have an inverse over its entire domain?

$g(x)$ is always between -1 and 1.

The function is continuous everywhere.

If a function fails the horizontal line test, what must be true about the function?

The function is not one-to-one.

The function is not continuous.

The function does not pass the vertical line test.

The domain of the function is restricted.

Inverse Functions Practice — Grade 4 amc_math

Lesson 16.4: Inverse Functions — Practice Questions

1. Why does the function $f(x) = x^2 + 5$ not have an inverse over the set of all real numbers?
- A. The function's graph does not have a y-intercept.
- B. There are at least two different inputs that produce the same output.
- C. The function is not defined for all possible input values.
- D. The output of the function is always positive.
2. The function $h(x) = |x - 2|$ is not one-to-one. Given that $h(5) = 3$, find a different value $c$ such that $c \neq 5$ and $h(c) = 3$. $c = \_\_\_$
3. Which statement best explains why the function $g(x) = \cos(x)$ does not have an inverse over its entire domain?
- A. $g(0) = 1$ and $g(\pi) = -1$
- B. The function is periodic.
- C. $g(x)$ is always between -1 and 1.
- D. The function is continuous everywhere.
4. The function $f(x) = x^2 - 10$ is not invertible on the domain of all real numbers. Given that $f(4) = 6$, find a different input value $x$ for which $f(x) = 6$. $x = \_\_\_$
5. If a function fails the horizontal line test, what must be true about the function?
- A. The function is not one-to-one.
- B. The function is not continuous.
- C. The function does not pass the vertical line test.
- D. The domain of the function is restricted.
6. Which of the following pairs of functions are inverses of each other, meaning one operation undoes the other?
- A. $f(x) = x + 5$ and $g(x) = 5x$
- B. $f(x) = 4x$ and $g(x) = x - 4$
- C. $f(x) = \frac{x}{2}$ and $g(x) = 2x$
- D. $f(x) = x^2$ and $g(x) = 2x$
7. If a function is defined as $f(x) = 4x + 5$, find its inverse function, $f^{-1}(x)$. $f^{-1}(x) = $ ___
8. What is the inverse function of $f(x) = x^4$?
- A. $g(x) = 4x$
- B. $g(x) = x - 4$
- C. $g(x) = \sqrt[4]{x}$
- D. $g(x) = \frac{x}{4}$
9. The functions $f(x) = x - 10$ and $g(x) = x + 10$ are inverses. Based on the property of inverse functions, what is the value of $f(g(15))$? The value is ___.
10. Find the inverse function, $g(x)$, for the function $f(x) = \frac{x-2}{5}$.
- A. $g(x) = 5x + 2$
- B. $g(x) = 5x - 2$
- C. $g(x) = \frac{x+2}{5}$
- D. $g(x) = 2x + 5$