Question

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

(1 point)
Responses

The graph of the inverse of f(x)
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)
𝑓
(
𝑥
)
is not one-to-one; thus, the inverse of f−1(x)
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)
𝑓
(
𝑥
)
passes the horizontal line test.
The function f left parenthesis x right parenthesis passes the horizontal line test.

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

Answer 1

To determine the correct statement regarding the function \( f(x) = 4x^4 - 1 \), let’s 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.

2. **Inverse Function**: If a function is not one-to-one, its inverse will not be a function.

### Step 1: Analyze the Function
The given function \( f(x) = 4x^4 - 1 \) is a polynomial of degree 4. The form of this polynomial indicates that it has a minimum point (since it is an even-degree polynomial with a positive leading coefficient).

### Step 2: Test for One-to-One
To check if it passes the horizontal line test, you can consider the derivative:

\[
f'(x) = 16x^3
\]

The derivative equals zero when \( x = 0 \).

- For \( x < 0 \), \( f'(x) < 0 \) (function is decreasing).
- For \( x > 0 \), \( f'(x) > 0 \) (function is increasing).

Since the function decreases to a minimum at \( x = 0 \) and then increases, it implies the function is not one-to-one (the same value is achieved at different points, e.g., both \( f(-a) \) and \( f(a) \) result in the same value).

### Step 3: Conclusion about Inverses
Since \( f(x) \) is not one-to-one, it does not pass the horizontal line test, and therefore, it does not have an inverse that is a function.

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