What function is the inverse of the function $$ ? $$ $$ f(x)=11 x^{2} $$ A. $$ s(x)= \pm \frac{\sqrt{11 x}}{x} $$, for $$ x0 $$ B. $$ q(x)= \pm \frac{\sqrt{11 x}}{11} $$, for $$ x \geq 0 $$ C. $$ r(x)= \pm \frac{\sqrt{x}}{11} $$, for $$ x \geq 0 $$ D. $$ p(x)= \pm \sqrt{11 x} $$, for $$ x \geq 0 $$

Question

What function is the inverse of the function $$ ? $$ $$ f(x)=11 x^{2} $$ A. $$ s(x)= \pm \frac{\sqrt{11 x}}{x} $$, for $$ x>0 $$ B. $$ q(x)= \pm \frac{\sqrt{11 x}}{11} $$, for $$ x \geq 0 $$ C. $$ r(x)= \pm \frac{\sqrt{x}}{11} $$, for $$ x \geq 0 $$ D. $$ p(x)= \pm \sqrt{11 x} $$, for $$ x \geq 0 $$

UpStudy AI · Accepted Answer

To find the inverse of the function $$ f(x) = 11x^2 $$, we need to follow these steps:

1. **Express the function in terms of $$ y $$:**
   $$
   y = 11x^2
   $$

2. **Solve for $$ x $$:**
   $$
   x^2 = \frac{y}{11}
   $$
   $$
   x = \pm \sqrt{\frac{y}{11}} = \pm \frac{\sqrt{y}}{\sqrt{11}} = \pm \frac{\sqrt{11y}}{11}
   $$

3. **Define the inverse function:**
   Since $$ f(x) = 11x^2 $$ is not one-to-one over all real numbers, we typically restrict the domain to $$ x \geq 0 $$ to make it invertible. Thus, the inverse function is:
   $$
   f^{-1}(y) = \pm \frac{\sqrt{11y}}{11}, \quad \text{for } y \geq 0
   $$

Looking at the provided options:

- **Option B:** $$ q(x) = \pm \frac{\sqrt{11x}}{11} $$, for $$ x \geq 0 $$

This matches our derived inverse function exactly.

**Answer:**

**Option B**

$$ q(x)= \pm \frac{\sqrt{11 x}}{11} $$, for $$ x \geq 0 $$
The inverse function is $$ q(x) = \pm \frac{\sqrt{11 x}}{11} $$, for $$ x \geq 0 $$.

What function is the inverse of the function \( ? \) \[ f(x)=11 x^{2} \] A. \( s(x)= \pm \frac{\sqrt{11 x}}{x} \), for \( x>0 \) B. \( q(x)= \pm \frac{\sqrt{11 x}}{11} \), for \( x \geq 0 \) C. \( r(x)= \pm \frac{\sqrt{x}}{11} \), for \( x \geq 0 \) D. \( p(x)= \pm \sqrt{11 x} \), for \( x \geq 0 \)

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