To find the inverse of the function [tex]\( f(x) = -\frac{1}{7} \sqrt{16 - x^2} \)[/tex] and determine whether the inverse is itself a function, follow these steps:
1. Start with the equation of the function:
[tex]\[
f(x) = -\frac{1}{7} \sqrt{16 - x^2}
\][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{7} \sqrt{16 - x^2}
\][/tex]
3. Isolate the square root term:
Negate both sides of the equation to remove the negative sign and multiply both sides by 7 to clear the fraction.
[tex]\[
-y = \frac{1}{7} \sqrt{16 - x^2}
\][/tex]
[tex]\[
-7y = \sqrt{16 - x^2}
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
(-7y)^2 = 16 - x^2
\][/tex]
[tex]\[
49y^2 = 16 - x^2
\][/tex]
5. Rearrange the equation to solve for [tex]\( x^2 \)[/tex]:
Add [tex]\( x^2 \)[/tex] to both sides of the equation and then subtract [tex]\( 49y^2 \)[/tex] from both sides:
[tex]\[
16 = x^2 + 49y^2
\][/tex]
[tex]\[
x^2 = 16 - 49y^2
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides:
[tex]\[
x = \pm \sqrt{16 - 49y^2}
\][/tex]
7. Determine the inverse function:
At this point, we see that there are potentially two solutions for [tex]\( x \)[/tex] given [tex]\( y \)[/tex]:
[tex]\[
x = \sqrt{16 - 49y^2} \quad \text{and} \quad x = -\sqrt{16 - 49y^2}
\][/tex]
Since there are two solutions, the inverse is not a single-valued function. Therefore, the inverse of [tex]\( f(x) \)[/tex] is not a function because the original function does not pass the horizontal line test (it's not one-to-one).
To summarize:
- The expression obtained for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] are two-valued:
[tex]\[
x = \sqrt{16 - 49y^2} \quad \text{and} \quad x = -\sqrt{16 - 49y^2}
\][/tex]
- Since there are two possible values for [tex]\( x \)[/tex] for each [tex]\( y \)[/tex], the result shows that [tex]\( f^{-1}(x) \)[/tex] is not a function.
Thus, the inverse [tex]\( f^{-1}(x) \)[/tex] does not exist in the traditional sense as a function.
