Problem 10 - 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 . \(f(x)=3-4 x, \quad g(x)=\frac{3-x}{4}\)

Step-by-Step Solution

Answer

Both the definition method and the graphing method confirm that \(g(x)\) and \(f(x)\) are inverse functions. As well, testing a few points supports this conclusion.

Step 1: Use the definition method

This method involves showing that when one function is applied to the result of other, you get the original input. Start by applying g to f(x), thus g(f(x)), and simplify the result. G(f(x)) can be evaluated by substituting \(f(x) = 3 - 4x\) into \(g(x) = \frac{3 - x}{4}\). After simplifying you get \(g(f(x)) = x\). Next, apply f to g(x), noted as f(g(x)), and simplify the result. \(f(g(x)) = f\left(\frac{3 - x}{4}\right)\). Simplifying gives \(f(g(x)) = x\).

Step 2: Use the graphing method

The idea of this method is to show that the two graphs are mirror images of each other across the line y=x. Hence, graph the lines y = f(x) and y = g(x) as well as the line y=x. Checking visually displays that \(f(x) = 3 - 4x\) is a downward sloping line which intersects the y-axis at \(y = 3\). The line \(g(x) = \frac{3 - x}{4}\) is an upward-sloping line which also intersects the y-axis at \(y = 3\). These two lines are reflections of each other around the line \(y=x\).

Step 3: Test a few points

Let's choose some points to reiterate the results from steps 1 and 2. For instance, choose \(x = 0\) for \(f(x)\) and \(g(x)\). Evaluating gives \(f(0) = 3\) and \(g(0) = 0.75\). Substituting these values into the other function, you'll get the original input, \(x = 0\) for both functions, supporting that they are inverse. You can try with more points.

Key Concepts

Function Composition

When evaluating if two functions are inverses, function composition plays a key role.
It involves applying one function to the output of another function. If two functions are inverse to each other, then composing them returns the original input.
Let's see this in action with our functions, \(f(x) = 3 - 4x\) and \(g(x) = \frac{3-x}{4}\). To check this

• First, compute \(g(f(x))\). Substitute \(f(x)\) in \(g(x)\), getting \(g(3 - 4x) = \frac{3 - (3 - 4x)}{4}\). Simplifying, you will find \(g(f(x)) = x\).
• Next, compute \(f(g(x))\). Substitute \(g(x)\) in \(f(x)\), which gives \(f\left(\frac{3 - x}{4}\right) = 3 - 4\left(\frac{3 - x}{4}\right)\). Simplify to get \(f(g(x)) = x\).

Both compositions result in \(x\), thus proving \(f\) and \(g\) are inverse functions through function composition.

Graphing Functions

Graphing functions provides a visual confirmation of whether two functions are inverses too.
If they are inverse, their graphs will be reflections across the line \(y = x\).
To graph our functions:

• For \(f(x) = 3 - 4x\), plot this line. It's a straight line with a negative slope intersecting the y-axis at \(y = 3\).
• For \(g(x) = \frac{3 - x}{4}\), plot it as well. This line has a positive slope and also intersects the y-axis at \(y = 3\).

The line \(y = x\) runs diagonally across the graph. You will notice these two function graphs, \(f(x)\) and \(g(x)\), are mirror images about the line \(y = x\). This visual confirmation strengthens our earlier finding that \(f(x)\) and \(g(x)\) are inverses.

Reflections

The concept of reflections in mathematics describes how one shape can be flipped over a line to coincide with another.
This is integral when dealing with inverse functions. With the reflection property, a function's graph and its inverse will look like mirror images over the line \(y=x\).
Consider the functions \(f(x) = 3 - 4x\) and \(g(x) = \frac{3-x}{4}\):

• The line \(y = x\) acts as our 'mirror'.
• Reflecting \(f(x)\) over \(y = x\) results in a line identical to the graph of \(g(x)\). Similarly, reflecting \(g(x)\) over \(y = x\) gives us the graph of \(f(x)\).

Thus, reflections not only provide a geometric insight into inverses, but they also solidify the relationship between them. Understanding how this reflection works makes it easier to identify inverse functions using their geometric properties.