Find the inverse of $f$. Then sketch the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes.
$$f ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0$$
A) $f^{-1}(x)=\sqrt{4+x^{2}}, x \geq 0$
B)$f ^ { - 1 } ( x ) = - \sqrt { 16 - x ^ { 2 } } , x \geq 0$
C)$f^{-1}(x)=\sqrt{16-x^{2}}, x \geq 0$
D) $f ^ { - 1 } ( x ) = - \sqrt { 4 + x ^ { 2 } } , x \geq 0$

Question

Find the inverse of $f$. Then sketch the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes.
$$f ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0$$
A) $f^{-1}(x)=\sqrt{4+x^{2}}, x \geq 0$ 
B)$f ^ { - 1 } ( x ) = - \sqrt { 16 - x ^ { 2 } } , x \geq 0$ 
C)$f^{-1}(x)=\sqrt{16-x^{2}}, x \geq 0$ 
D) $f ^ { - 1 } ( x ) = - \sqrt { 4 + x ^ { 2 } } , x \geq 0$

Quizplus · Accepted Answer

To find the inverse of $f$, we switch the roles of $x$ and $y$:
$$x = \sqrt{16 - y^2}$$
Then we solve for $y$:
$$y = \sqrt{16 - x^2}$$
This gives us $f ^ { - 1 } ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0$, which is choice C.

To sketch the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes, we first note that the domain and range of $f$ are both $[0, 4]$. The graph of $f$ is half of a circle with radius 4 centered at the origin, restricted to the first quadrant. The graph of $f ^ { - 1 }$ is obtained by reflecting this half-circle across the line $y = x$, which is the line of symmetry.

Therefore, the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes look like this:

11ecd741_ddc4_08f1_90c3_990e2ff314c7_TB8834_11

Find the Inverse Of fff Then Sketch the Graphs Of Then

Find the Inverse Of $f$ Then Sketch the Graphs Of Then