Question

Use the image to answer the question.

A parabola is graphed on a coordinate plane. The x-axis ranges from negative 5 to 5 in increments of 1. The y-axis ranges from negative 2 to 8 in increments of 1.

Which of the following statements best describes the function f(x)=4x4−1
?

(1 point)
Responses

The function f(x)
has an inverse that is a function.
The function f left parenthesis x right parenthesis has an inverse that is a function.

The graph of the inverse of f(x)
passes the vertical line test.
The graph of the inverse of f left parenthesis x right parenthesis passes the vertical line test.

The function f(x)
is not one-to-one; thus, the inverse of f−1(x)
is not a function.
The function f left parenthesis x right parenthesis is not one-to-one; thus, the inverse of f inverse left parenthesis x right parenthesis is not a function.

The function f(x)
passes the horizontal line test.

Answer 1

To determine the best description of the function \( f(x) = 4x^4 - 1 \), we need to analyze its properties.

1. **One-to-One Function**: A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function more than once. Since \( x^4 \) is a polynomial function that is always non-negative for all real \( x \), the function \( f(x) = 4x^4 - 1 \) is increasing for \( x \geq 0 \) and decreasing for \( x < 0 \). Thus, there are values of \( x \) that yield the same \( f(x) \) value for different \( x \) inputs (for example, \( f(-1) = f(1) \)). Therefore, \( f(x) \) does not pass the horizontal line test and is not one-to-one.

2. **Inverse**: Since \( f(x) \) is not one-to-one, its inverse \( f^{-1}(x) \) is not a function.

Based on this analysis, the most accurate statement is:

- **The function \( f(x) \) is not one-to-one; thus, the inverse of \( f^{-1}(x) \) is not a function.**

This option clearly states that the lack of one-to-one property results in the inverse not being a function.