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Chapter 5: Problem 6

Show that \(f\) and \(g\) are inverse functions (a) analytically and (b) graphically. \(f(x)=16-x^{2}, \quad x \geq 0, \quad g(x)=\sqrt{16-x}\)

Short Answer

Expert verified
Analytically, both the composite functions \(f(g(x))\) and \(g(f(x))\) equal to \(x\), showing that \(f\) and \(g\) are inverses of each other. Graphically, the graph of \(g(x)\) is a reflection of the graph of \(f(x)\) over the line \(y=x\), which also shows that \(f\) and \(g\) are inverses of each other.

Step by step solution

01

Analytical Proof: Step 1

First, let's find \(f(g(x))\). We plug \(g(x)\) into \(f(x)\), so we get: \(f(g(x)) = f(\sqrt{16-x})\). Expanding, this gives us: \(f(\sqrt{16-x}) = 16 - (\sqrt{16-x})^{2}\).
02

Analytical Proof: Step 2

Since \((\sqrt{16-x})^{2}\) is simply \(16-x\), the expression simplifies to: \(f(g(x)) = 16 - (16 - x) = x\). Thus, \(f(g(x))=x\), which is one half of the condition for \(f\) and \(g\) to be inverse functions.
03

Analytical Proof: Step 3

Now let's find \(g(f(x))\). We plug \(f(x)\) into \(g(x)\), to get: \(g(f(x)) = g(16-x^{2})\). Expanding, this gives us: \(g(16-x^{2}) = \sqrt{16 - (16 - x^{2})}\).
04

Analytical Proof: Step 4

The expression under the square root simplifies to \(x^{2}\), so our expression now is: \(g(f(x)) = \sqrt{x^2}\). Since \(x \geq 0\), \(\sqrt{x^{2}}\) equals \(x\) and our current expression simplifies to: \(g(f(x)) = x\). Therefore, both \(f(g(x))=x\) and \(g(f(x))=x\), satisfying the conditions for \(f\) and \(g\) to be inverses of each other analytically.
05

Graphical Proof: Step 1

To prove the functions are inverses graphically, start by sketching the graph of \(y=f(x)\). This is a downward-opening parabola centered at \(x=0\) and with a vertex at (0,16). The restriction on \(x \geq 0\) makes it half of a full parabola.
06

Graphical Proof: Step 2

Sketch the graph of \(y=g(x)\). This is a sideways-opening parabola (actually half of one since the range is restricted to \(y \geq 0\)). It opens to the right and is centered on \(y=0\) with a vertex at (16,0).
07

Graphical Proof: Step 3

Sketch the line \(y=x\), which is the line of reflection for inverse functions. If \(f(x)\) and \(g(x)\) are indeed inverses, the reflection of \(f(x)\) over \(y=x\) should match the graph of \(g(x)\). Once you see that this is the case, you will have completed the proof graphically.

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