Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Show that \(f\) and \(g\) are inverse functions by (a) using the definition of inverse functions and (b) graphing the functions. Make sure you test a few points, as shown in Examples 6 and 7 . \(f(x)=5 x+1, \quad g(x)=\frac{x-1}{5}\)

Step-by-Step Solution

Answer

Function \(f(x) = 5x + 1\) and function \(g(x) = \frac{x - 1}{5}\) are indeed inverses of each other. We've verified this by substituting \(g(x)\) into \(f(x)\) and \(f(x)\) into \(g(x)\), and both resulted in \(x\). The graphs of \(f\) and \(g\) are mirror images over the line \(y = x\), verifying they are inverse functions graphically.

Step 1: Applying the Definition of Inverse Functions

Inverse functions, by definition, should satisfy the following conditions: \(f(g(x)) = x\) and \(g(f(x)) = x\). So, start by substituting \(g(x)\) into function \(f\) and vice versa.

Step 2: Verify \(f(g(x)) = x\)

With \(g(x)\) substituted into \(f\), \(f(g(x)) = f(\frac{x-1}{5}) = 5(\frac{x-1}{5}) + 1 = x - 1 + 1 = x\). As such, the condition \(f(g(x)) = x\) is satisfied.

Step 3: Verify \(g(f(x)) = x\)

Now, substitute \(f(x)\) into \(g\), \(g(f(x)) = g(5x+1) = \frac{5x+1-1}{5} = x\). The condition \(g(f(x)) = x\) is also satisfied.

Step 4: Graphing the Functions

Next, plot both functions \(f(x)\) and \(g(x)\) on the same graph. Then draw the line \(y = x\). Because they are inverse functions, the graphs of \(f\) and \(g\) should be reflections over the line \(y=x\). However, it is advised to try some points for clearer understanding.

Step 5: Testing Points

Let's choose a few points and plot them both on the \(f\) and \(g\) functions. As examples, we can use \(x = -1, 0, 1\). Then compare these points with their counterparts on the other function's graph to confirm the reflection.

Key Concepts

Function Composition

Function composition is a way to build complex functions by combining simpler ones. When we have two functions, say \( f \) and \( g \), the function composition \( f(g(x)) \) is formed by sending the input \( x \) through function \( g \) first, and then taking the output of that through function \( f \). This is a cornerstone of checking whether two functions are inverses.

For inverse functions, you need this composition to return the same value as the original input, meaning that \( f(g(x)) = x \) and \( g(f(x)) = x \). These conditions ensure that each function undoes the effect of the other. In the given exercise, \( f(x) = 5x+1 \) and \( g(x) = \frac{x-1}{5} \), confirm their inverse nature through composition:

• For \( f(g(x)) = f\left(\frac{x-1}{5}\right) = x \), meaning that \( f \) correctly cancels out the transformation by \( g \).
• Similarly, \( g(f(x)) = g(5x+1) = x \), demonstrating that \( g \) also reverses the transformation by \( f \).

Function composition helps us ensure that function pairs truly work as each other's reverse.

Reflection Over the Line y=x

When two functions are inverses, their graphs have a special property: they reflect over the line \( y=x \). This means that every point \((a, b)\) on the graph of \( f \) corresponds to a point \((b, a)\) on the graph of \( g \), and vice versa. This is a visual way to verify the inverse relationship.

In the exercise, plotting \( f(x) = 5x + 1 \) and \( g(x) = \frac{x - 1}{5} \) on the coordinate plane and drawing the line \( y = x \) between them shows this reflection property clearly. By using test points like \( x = -1, 0, 1 \), you can see how points on one graph translate to the points on the other. For instance:

• \((-1, -4)\) on \( f(x) \) maps to \((-4, -1)\) on \( g(x) \).
• \((0, 1)\) on \( f(x) \) corresponds to \((1, 0)\) on \( g(x) \).
• \((1, 6)\) on \( f(x) \) corresponds to \((6, 1)\) on \( g(x) \).

Testing with these points gives you confidence in their inverse relationship. Reflecting over the line \( y=x \) is a practical method to confirm inverses by using graphical representation.

Graphing Functions

Graphing functions is an effective way to visualize mathematical relationships. By plotting both \( f(x) \) and \( g(x) \) on a graph, you can see their inverse relationship graphically.

Start by drawing each function individually:

• The line for \( f(x) = 5x + 1 \) is a straight line with a slope of 5 and a y-intercept at 1.
• For \( g(x) = \frac{x-1}{5} \), you have a line with a much smaller slope of \( \frac{1}{5} \) and a y-intercept at \( -\frac{1}{5} \).

Next, draw the line \( y = x \), which acts as the mirror line for function reflection. Check if every point on \( f(x) \) mirrors to \( g(x) \).

Graphing provides:

• A visual confirmation of the inverse property through reflection.
• Understanding of how inverses function and what it looks like geometrically.

This not only tests the inverse relationship but also deepens your comprehension of function behavior.