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Chapter 1: Problem 27

Verify that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. $$f(x)=\sqrt{x-4}, \quad g(x)=x^{2}+4, \quad x \geq 0$$

Short Answer

Expert verified
Therefore, functions \(f\) and \(g\) are inverse function to each other which is verified both algebraically and graphically.

Step by step solution

01

- Algebraic Verification

To verify algebraically whether \(f\) and \(g\) are inverse of each other, apply function \(g\) into function \(f\) and function \(f\) into function \(g\) to check if \(f(g(x))\) and \(g(f(x))\), both are equal to \(x\).Compute \(f(g(x)) = f(x^{2} + 4)\) and simplify to see if the result is \(x\). Then, compute \(g(f(x)) = g(\sqrt{x - 4})\) and simplify to see if the result is \(x\).
02

- Computation of f(g(x))

Substitute \(x^{2} + 4\) into \(f(x)\):\(f(g(x)) = f(x^{2} + 4) = \sqrt{x^{2} + 4 - 4} = \sqrt{x^{2}}\). Since \(x \geq 0\), \(\sqrt{x^{2}}\) simplifies to \(x\).
03

- Computation of g(f(x))

Substitute \(\sqrt{x - 4}\) into \(g(x)\):\(g(f(x)) = g(\sqrt{x - 4}) = (\sqrt{x - 4})^{2} + 4 = x - 4 + 4 = x\).
04

- Graphical Verification

To check graphically, plot the graphs of \(f(x)\) and \(g(x)\), and the line \(y = x\). If \(f\) and \(g\) are inverses, then the graphs of \(f(x)\) and \(g(x)\) should be reflections of each other across the line \(y = x\).

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