Problem 24 - Precalculus with Limits

Question

In Exercises 23-34, show that \(f\) and \(g\) are inverse functions (a) algebraically and (b) graphically. \(f(x) = x - 5\), \(g(x) = x + 5\)

Step-by-Step Solution

Answer

The functions \(f(x) = x - 5\) and \(g(x) = x + 5\) are inverse functions to each other because both \(f(g(x))\) and \(g(f(x))\) return \(x\), and their graphs are reflections of each other across the line \(y = x\).

Step 1: Algebraic Solution

Compute \(f(g(x))\) and \(g(f(x))\). If \(f\) and \(g\) are inverse functions, these both should equal to \(x\). Let's calculate it:\n\na) Evaluate \(f(g(x)) = f(x + 5) = (x + 5) - 5 = x\).\nb) Evaluate \(g(f(x)) = g(x - 5) = (x - 5) + 5 = x\).\n\nTherefore, \(f\) and \(g\) are inverse functions as both \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 2: Graphical Solution

To show this graphically, plot the graph of \(y = f(x)\) and \(y = g(x)\). Also plot the line \(y = x\) for reference. The graph of \(f(x)\) will be a line with a slope of 1 and y-intercept at -5. The graph of \(g(x)\) will also be a line with a slope of 1 and a y-intercept of 5. These two graphs should be reflections of each other across the line \(y = x\), which means that \(f\) and \(g\) are inverse functions.

Key Concepts

Algebraic Proof

When proving whether two functions are inverses of each other algebraically, the main idea is to check if applying one function after the other gets you back to your starting point. In simpler terms, for two functions \(f\) and \(g\), they are inverses if and only if \(f(g(x)) = x\) and \(g(f(x)) = x\). This shows that the functions cancel each other out, leaving you with \(x\). Here’s how it works for our example:

• First, calculate \(f(g(x)) = f(x + 5) = (x + 5) - 5\). The \(+5\) and \(-5\) cancel out, so you’re left with \(x\).


• Next, calculate \(g(f(x)) = g(x - 5) = (x - 5) + 5\). Similarly, the \(-5\) and \(+5\) cancel each other, resulting in \(x\).

Both expressions give us \(x\), confirming that \(f\) and \(g\) are indeed inverse functions.

Graphical Representation

Graphing is a visual way to confirm that two functions are inverses. If \(f\) and \(g\) are inverse functions, their graphs will be reflections of one another across the line \(y = x\). This line acts like a mirror.

• For our functions \(f(x) = x - 5\) and \(g(x) = x + 5\), begin by plotting them on a coordinate plane.


• Graph \(y = f(x)\), which is a straight line with a slope of 1 and a y-intercept at -5. This line will go through points like \((-5, -10)\), \((0, -5)\), and \((5, 0)\).
• Next, plot \(y = g(x)\), also a line with a slope of 1 but with a y-intercept at 5. It will touch points like \((-5, 0)\), \((0, 5)\), and \((5, 10)\).

By reflecting over the line \(y = x\), each corresponding point of \(f(x)\) and \(g(x)\) matches exactly, reinforcing that they are inverse functions.

Functions and Their Inverses

Understanding functions and their inverses uncovers a key relationship in mathematics. Functions map inputs to outputs, kind of like a precise recipe where each ingredient gives you specific results. Inverse functions essentially flip this process around.

• When you have a function \(f(x)\), its inverse \(g(x)\) will reverse this transformation.


• Algebraically, if you start with a value, apply \(f\), and then apply \(g\), you should end up back at your original value, just as we demonstrated with \(f(g(x)) = x\) and \(g(f(x)) = x\).
• Graphically, the interplay between functions and their inverses revolves around symmetry about the line \(y = x\). Seeing two lines, such as \(f(x)\) and \(g(x)\), mirror each other around this line confirms their inverse relationship.

Linear Functions

Linear functions form the simplest category of functions, represented mathematically by the formula \(y = mx + b\). The beauty of linear functions lies in their constant rate of change, evidenced by their slope \(m\) and straight-line graph.

• In our example, \(f(x) = x - 5\) and \(g(x) = x + 5\) are linear functions where the slope \(m\) is 1, exemplifying a clean, straight-path change.


• Linear functions are intuitive to understand and can transform into their inverses by adjusting their y-intercepts.
• When a linear function such as \(y = x - 5\) is inversed to become \(y = x + 5\), the slope stays the same, but the line shifts, reflecting across \(y = x\). This shift in the intercept by equal and opposite amounts is typical in linear inverse functions.

Therefore, grasping linear functions sets a solid foundation to delve deeper into more complex functions and their inverses in algebra.