Question

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

Functions \(f(x) = 16 - x^2\) and \(g(x) = \sqrt{16 - x}\) are inverse functions. Analytically, it is shown that \(f(g(x)) = x\) and \(g(f(x)) = x\), while graphically their mirror image relationship about the line \(y = x\) confirms them as inverses.

Step-by-step solution

Analytically Prove \(f(g(x)) = x\)

First, substitute \(g(x)\) into \(f(x)\) to get \(f(g(x))\). \nTherefore, \(f(g(x)) = f(\sqrt{16-x}) = 16 - (\sqrt{16 - x})^2\), which simplifies to \(f(g(x)) = x\). This proves \(f(g(x)) = x\)

Analytically Prove \(g(f(x)) = x\)

Second, substitute \(f(x)\) into \(g(x)\) to get \(g(f(x))\). \nSo, \(g(f(x)) = g(16-x^2) = \sqrt{16 - (16 - x^2)}\), which simplifies to \(g(f(x)) = x\). This proves \(g(f(x)) = x\)

Graphically Show That \(f\) and \(g\) Are Inverses

Plot \(f(x)\) and \(g(x)\) on the same graph. \(f(x)\) is a descending parabola starting from 16 at \(x = 0\) while \(g(x)\) is a square root function starting from \(x = 0\). If these graphs are mirror images of each other about the line \(y = x\), then they are inverse functions. The graphical representation verifies that \(f\) and \(g\) are indeed inverses.

Key concepts

Analytical Proof

Understanding the nuances of inverse functions is a critical component of advanced algebra. An analytical proof directly tackles the underlying mathematics.
In this particular exercise, we are given two functions, \( f(x) = 16 - x^2 \) with \( x \geq 0 \) and \( g(x) = \sqrt{16-x} \). To analytically prove that \( f \) and \( g \) are inverses, we must show that when we perform function composition—inserting one function into the other—we retrieve the original input value.

Performing this operation, we first consider \( f(g(x)) \). Substituting \( g(x) \) into \( f(x) \), we have: \[ f(g(x)) = f(\sqrt{16-x}) = 16 - (\sqrt{16 - x})^2 \] Simplifying the expression, the square and the square root cancel each other, thus yielding \( f(g(x)) = x \). Next, we must prove that \( g(f(x)) = x \) as well. Substituting \( f(x) \) into \( g(x) \), we get: \[ g(f(x)) = g(16-x^2) = \sqrt{16 - (16 - x^2)} \] After simplification, this too confirms \( g(f(x)) = x \). The perfect reciprocity of these operations offers a clear analytical proof that \( f \) and \( g \) are indeed inverse functions.

Graphical Proof

The graphical proof provides a visual confirmation of the relationship between inverse functions. Graphs aid in deeper comprehension and often reveal properties that might not be immediately evident from equations alone.

To graphically demonstrate that \( f \) and \( g \) are inverse functions, we plot both functions on the same coordinate plane. The function \( f(x) = 16 - x^2 \) is a parabola opening downwards with its vertex at the point \( (0, 16) \). Conversely, the function \( g(x) = \sqrt{16-x} \) is the top half of a sideways parabola with its highest point at the same vertex.
When we look at these graphs, the tell-tale sign that two functions are inverses is if one is the mirror image of the other across the line \( y = x \). In our case, if we were to reflect the graph of \( f(x) \) over the line \( y = x \), it should overlap perfectly with the graph of \( g(x) \). This graphical symmetry reinforces the idea that applying one function after the other returns the input value, thus illustrating their inverse relationship.

Function Composition

At its core, function composition is about applying functions in succession; that is, the output of one function becomes the input of another. For functions \( f \) and \( g \) to be considered inverses, each must undo the work of the other, leading us back to our starting value.

This interplay can be formally expressed as \( f(g(x)) = x \) and \( g(f(x)) = x \). When we graph these composite functions, they should form straight lines, with slopes that are multiplicative inverses of each other. This concept not only helps identify inverse functions but also is vital in solving complex equations and transforming coordinate points across various functions.

By understanding how to manipulate and combine functions through composition, students can solve a diverse array of mathematical challenges. It is one of the basic building blocks of the study of functions, making it a fundamental concept that goes far beyond just proving that two functions are inverses.