Question

Question was provided via image upload.

Explanation:

Identify the given function

We are given the rational function:

$$f(x) = \frac{1}{3 + x^2}$$

Analyze the incomplete prompt

The prompt asks to "find the corresponding \(x\)". Since the value of \(f(x)\) (or \(y\)) is not specified in the image, we will solve for \(x\) in terms of \(y\), where \(y = f(x)\).

Set up the equation

Let \(y = f(x)\). We write the equation as:

$$y = \frac{1}{3 + x^2}$$

Rearrange to solve for x

Multiply both sides by \(3 + x^2\) (since \(3 + x^2 \ge 3 > 0\)):

$$y(3 + x^2) = 1$$

$$3y + yx^2 = 1$$

$$yx^2 = 1 - 3y$$

Isolate x

Divide by \(y\) (assuming \(y
eq 0\)):

$$x^2 = \frac{1 - 3y}{y}$$

Taking the square root of both sides gives:

$$x = \pm\sqrt{\frac{1 - 3y}{y}}$$

This solution is valid for \(0 < y \le \frac{1}{3}\).

Answer:

$$x = \pm\sqrt{\frac{1 - f(x)}{f(x)}}$$

(Note: Since the specific value for \(f(x)\) is cut off in the image, the general solution expressing \(x\) in terms of \(f(x)\) is provided.)