Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=4 x+3, \quad g(x)=\frac{x-3}{4}$$

Step-by-Step Solution

Answer

\(f(x)\) and \(g(x)\) are inverses of each other since \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Understand the Definition of Inverse Functions

Two functions, \(f\) and \(g\), are inverses of each other if \(f(g(x)) = x\) and \(g(f(x)) = x\) for every \(x\) in the domains of \(g\) and \(f\), respectively. This means that applying \(g\) to \(f(x)\) should yield \(x\), and applying \(f\) to \(g(x)\) should also yield \(x\).

Step 2: Compute \(f(g(x))\)

Substitute \(g(x)\) into \(f(x)\). We have \(g(x) = \frac{x-3}{4}\). Substitute this into \(f(x) = 4x + 3\):\[f(g(x)) = f\left(\frac{x-3}{4}\right) = 4\left(\frac{x-3}{4}\right) + 3.\]Simplify the expression:\[f(g(x)) = (x-3) + 3 = x.\]

Step 3: Compute \(g(f(x))\)

Substitute \(f(x)\) into \(g(x)\). We have \(f(x) = 4x + 3\). Substitute this into \(g(x) = \frac{x-3}{4}\):\[g(f(x)) = g(4x + 3) = \frac{(4x + 3)-3}{4}.\]Simplify the expression:\[g(f(x)) = \frac{4x}{4} = x.\]

Step 4: Confirm that \(f\) and \(g\) Are Inverses

Since we have shown that \(f(g(x)) = x\) and \(g(f(x)) = x\), \(f\) and \(g\) are indeed inverses of each other by definition.

Key Concepts

Definition of Inverse Functions

When we talk about inverse functions, we're looking at two functions that essentially 'undo' each other. Imagine you have two functions, let's call them \( f \) and \( g \). These functions are inverses if when you plug \( g(x) \) into \( f(x) \), you get \( x \) back, and vice versa.

To understand this better, think of \( f \) as a machine that does something to an input \( x \), and \( g \) as a machine that reverses whatever \( f \) did, taking it back to the original \( x \). Mathematically, this is expressed as:

• \( f(g(x)) = x \)
• \( g(f(x)) = x \)

This condition must be true for every input \( x \) in the domains of their respective functions.

In simpler terms, if you start with a number, apply one function, and then the other, you end up back where you started. This property is what makes them inverses.

Composition of Functions

Composition of functions involves taking the output from one function and using it as the input for another. It's like stacking functions on top of each other, where one processes the output of another. This is represented as \( f(g(x)) \) or \( g(f(x)) \).

Let's break this down with our functions \( f(x) = 4x + 3 \) and \( g(x) = \frac{x-3}{4} \). When we say \( f(g(x)) \), we enter \( g(x) \) into \( f(x) \). So effectively:

• First, evaluate \( g(x) \): \( g(x) = \frac{x-3}{4} \)
• Then substitute \( g(x) \) into \( f(x) \): \( f(g(x)) = 4\left(\frac{x-3}{4}\right) + 3 \)
• Simplify: \( f(g(x)) = x \)

Similarly, for \( g(f(x)) \):

• Evaluate \( f(x) \): \( f(x) = 4x + 3 \)
• Then substitute \( f(x) \) into \( g(x) \): \( g(f(x)) = \frac{(4x + 3) - 3}{4} \)
• Simplify to get \( g(f(x)) = x \)

The result of these operations confirms that \( f \) and \( g \) are each other's inverses, as both compositions lead back to the original input \( x \).

Function Domain

The concept of a function's domain is crucial when dealing with inverse functions, as it determines the set of values that a function can accept as input. For a function \( f(x) \), the domain consists of all real numbers for which the function is defined.

When checking for inverses, the domain of \( f(x) \) needs to fit with what \( g(x) \) can accept, and vice versa. In our example, \( f(x) = 4x + 3 \) and \( g(x) = \frac{x-3}{4} \), both are defined for all real numbers, making their domains \( \mathbb{R} \).

This means that applying \( f \) and \( g \) to any real number will always yield a valid output, which is essential for them to function as inverses. The proper alignment of domains ensures the calculated inverses remain true for every \( x \) within those domains.