Problem 38 - A Graphical Approach to College Algebra

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

\(f\) and \(g\) are inverses as \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 1: Understand Inverse Functions

Two functions, \(f\) and \(g\), are inverses if \(f(g(x)) = x\) and \(g(f(x)) = x\) for all \(x\) in the domains of \(f\) and \(g\). We need to establish these identities for the given functions \(f(x) = 4x + 3\) and \(g(x) = \frac{x - 3}{4}\).

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

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

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

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

Step 4: Conclusion

Since both \(f(g(x)) = x\) and \(g(f(x)) = x\) are satisfied, \(f\) and \(g\) are inverse functions of each other by definition.

Key Concepts

Function Composition

Function composition is a process where you apply one function to the result of another function. This is similar to following a set of instructions step-by-step. Consider a function \(f(x)\) transforming \(x\) into a new value, like a recipe yielding a dish. When you compose functions, you're stacking processes: the outcome of \(g(x)\) becomes the input for \(f(x)\), or vice versa.

In the case of our exercise, we take \(g(x) = \frac{x - 3}{4}\) and feed it into \(f(x) = 4x + 3\), creating \(f(g(x))\). The key idea is to simplify \(f(g(x))\) until you get back \(x\), reinforcing that \(g\) undoes \(f\).

For example, the step \(f\left(\frac{x-3}{4}\right) = 4\left(\frac{x-3}{4}\right) + 3\) simplifies to \((x-3) + 3 = x\). This unraveling confirms that, through composition, functions can be re-ordered or reversed to reach the original input.

Algebra

Algebra is the language of expressing relationships using symbols and equations. Within the context of inverse functions, it's crucial to manipulate expressions to verify identities.

The transformative power of algebra shines when simplifying \(f(g(x))\) and \(g(f(x))\). Take for instance \(f(g(x)) = 4\left(\frac{x-3}{4}\right) + 3\), reduced to \((x-3)+3\), then to \(x\). This step-by-step simplification illustrates the exacting nature of algebra.

Similarly, the expression \(g(f(x)) = \frac{(4x + 3) - 3}{4}\) turns into \(\frac{4x}{4}\), simplifying finally to \(x\). Algebra empowers us to transform and simplify until the core identity emerges.

Analytical Verification

Analytical verification involves proving that two functions are inverses through systematic calculation. It’s like a mathematical detective work, ensuring each step logically leads to the conclusion.

In this task, we prove \(f(x) = 4x+3\) and \(g(x) = \frac{x-3}{4}\) are inverses by showing \(f(g(x)) = x\) and \(g(f(x)) = x\). This process involves substituting and simplifying to find consistency.

The substitutions show that both \(f(g(x)) = x\) and \(g(f(x)) = x\) simplify down to their respective identities. This means \(f\) and \(g\) effectively cancel each other out, no matter the input. This logical consistency is the backbone of proving inverse relationships.