Problem 37 - Algebra 2

Question

Determine whether each pair of functions are inverse functions. \(f(x)=4 x-5\) \(g(x)=\frac{1}{4} x-\frac{5}{16}\)

Step-by-Step Solution

Answer

The functions are not inverses, as \(f(g(x)) \neq x\).

Step 1: Understand Inverse Functions

Two functions, \(f(x)\) and \(g(x)\), are inverse functions if composing one with the other (in both orders) results in the identity function \(x\), i.e., \((f\circ g)(x) = x\) and \((g\circ f)(x) = x\).

Step 2: Compose f(g(x))

Substitute \(g(x)\) into \(f(x)\):\[ f(g(x)) = f\left( \frac{1}{4}x - \frac{5}{16} \right) = 4\left( \frac{1}{4}x - \frac{5}{16} \right) - 5 \]Calculate it:\[ = x - \frac{5}{4} - 5 = x - \frac{5}{4} - \frac{20}{4} = x - \frac{25}{4} \]Since it returns \(x - \frac{25}{4}\), this is not equal to \(x\).

Step 3: Conclude f and g are Not Inverses

Because \(f(g(x)) eq x\), the two functions cannot be inverse functions of each other. There's no need to calculate \(g(f(x))\) since both compositions need to result in \(x\) for them to be inverses.

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics, especially when dealing with inverse functions. It involves plugging one function into another, essentially creating a "function within a function". When you have two functions, say \( f(x) \) and \( g(x) \), composing them means substituting \( g(x) \) into \( f(x) \), indicated by \((f \circ g)(x)\). Similarly, you can substitute \( f(x) \) into \( g(x) \), denoted by \((g \circ f)(x)\).

This operation is crucial for determining if two functions are inverses of each other. For two functions to be inverses, both composed functions \((f \circ g)(x)\) and \((g \circ f)(x)\) must simplify to the identity function \(x\). If one of the compositions fails to simplify to \(x\), the functions are not inverses.

Always remember, composing functions is not as simple as multiplying two numbers. Each output of the first function becomes the input of the second. It can often feel like a puzzle, where you need to find whether the output matches the function’s end goal: \(x\).

Identity Function

The identity function is a unique concept in mathematics. It's like the "do-nothing" operation on numbers, represented simply by \(I(x) = x\). It outputs whatever is inputted without any modification.

In the realm of inverse functions, the identity function plays a pivotal role. When you compose two functions and each results in their own identity function, it means that both functions essentially "undo" each other. This undoing property is exactly what makes two functions inverses.

Layer after layer, if you keep applying the identity function, you won't alter the outcome. That’s its essential beauty. So, when working with function compositions to test for inverses, each resulting in \(x\) is a strong indicator they are true inverse pairs.

Algebraic Functions

Algebraic functions include a wide variety of functions formed using polynomial expressions. These often involve operations like addition, multiplication, division, and sometimes taking roots.

In the given exercise, both \(f(x) = 4x-5\) and \(g(x)=\frac{1}{4}x-\frac{5}{16}\) are algebraic functions. Understanding their form is key to seeing how they should interact when composed. These functions use linear expressions, which means they create straight lines when graphed, characterized by their slopes and intercepts.

To determine if they are inverses, you substitute \(g(x)\) into \(f(x)\) and attempt to simplify it to \(x\). The beauty of algebraic functions often lies in their predictability and structured form, making it straightforward to check compositions, though accuracy in algebraic manipulation is crucial to identify their relationships.