Determine whether the function has an inverse function. $$ f(x)=\frac{15}{x^{2}} $$ Yes, $$ f $$ does have an inverse. No, $$ f $$ does not have an inverse. If it does, then find the inverse function. (If an answer does not exist, enter DNE.) $$ f^{-1}(x)= $$

Question

UpStudy AI · Accepted Answer

To determine whether the function $$ f(x) = \frac{15}{x^2} $$ has an inverse, we need to check if it is a 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.

### Step 1: Analyze the function

The function $$ f(x) = \frac{15}{x^2} $$ is defined for all $$ x 
eq 0 $$.

- For $$ x > 0 $$, $$ f(x) $$ is positive and decreases as $$ x $$ increases.
- For $$ x < 0 $$, $$ f(x) $$ is also positive and decreases as $$ |x| $$ increases.

Since $$ f(x) $$ produces the same output for both positive and negative values of $$ x $$ (e.g., $$ f(2) = f(-2) = \frac{15}{4} $$), it fails the horizontal line test.

### Conclusion on Inverses

Since the function is not one-to-one, it does not have an inverse.

Thus, the answer is:
**No, $$ f $$ does not have an inverse.**

The inverse function $$ f^{-1}(x) $$ does not exist, so:
$$ f^{-1}(x) = 	ext{DNE} $$
No, $$ f $$ does not have an inverse.

Determine whether the function has an inverse function. \( f(x)=\frac{15}{x^{2}} \) Yes, \( f \) does have an inverse. No, \( f \) does not have an inverse. If it does, then find the inverse function. (If an answer does not exist, enter DNE.) \( f^{-1}(x)= \)

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Pre Calculus Questions