Question

In Exercises 49–52, determine whether the functions are inverses. $$ f(x)=7 x^{3 / 2}-4, g(x)=\left(\frac{x+4}{7}\right)^{3 / 2} $$

Short answer

The functions \(f(x)\) and \(g(x)\) are not inverses of each other as neither \(g(f(x))\) nor \(f(g(x))\) equals to x.

Step-by-step solution

Substitute \(f(x)\) into \(g(x)\)

We start by substituting the function \(f(x)=7 x^{3 / 2}-4\) into \(g(x)=\left(\frac{x+4}{7}\right)^{3 / 2}\). Thus, \(g(f(x))=\left(\frac{f(x)+4}{7}\right)^{2 / 3}\), which simplifies to \(\left(\frac{7x^{3/2} - 4+4}{7}\right)^{3 / 2} = (x^{3/2})^{3/2}\).

Simplify the resulting function

Simplify the resulting function to see if it equals to x. Using the power-of-a-power rule which states that \((a^m)^n = a^{mn}\), we have the function becomes \(x^{3/2*3/2} = x^{9/4}\).

Substitute \(g(x)\) into \(f(x)\)

Next, we substitute the function \(g(x)=\left(\frac{x+4}{7}\right)^{2 / 3}\) into \(f(x)=7 x^{3 / 2}-4\). The resulting function \(f(g(x))=7(g(x))^{3 / 2}-4 = 7\left(\frac{x+4}{7}\right)^2 - 4\) simplifies to \(7\left(\frac{x^2+16x+16}{49}\right) - 4\).

Simplify the resulting function

Simplify the resulting function to see if it equals to x. By multiplying out the terms, we get that this function simplifies to \((x^2+16x+16)/7 - 4\). This does not simplify to x.

Key concepts

Inverse of a Function

Understanding the inverse of a function is like learning how to reverse a journey. When you have a function, say, driving from your house to school, the inverse is the route from school back to your house. It's a way to undo what the original function did. Mathematically, if a function takes an input and gives you an output, its inverse will take that output and return the original input.

For a function to have an inverse, each input must be linked to a unique output, and each output must come from a unique input. This means, in functions with inverses, if you input different numbers, you should always get different results. We often note the inverse of a function f(x) as f^-1(x).

To check if two functions are inverses, you can perform a simple test: plug one function into the other and see if you get the identity function, that is, if f(g(x)) = x and g(f(x)) = x, where f and g are functions at hand. If both conditions are satisfied, f and g are indeed inverses of each other.

Power-of-a-Power Rule

The power-of-a-power rule is a foundational concept in algebra that helps us handle complex exponential expressions. When you have an exponent raised to another exponent, like a power tower, this rule comes into play. The rule states that \( (a^m)^n = a^{mn} \) - if you have a power raised to another power, you multiply the exponents.

Imagine you're stacking boxes, and in each box, there are smaller boxes of the same amount. If you have m boxes each containing n items, the total items you have is the product of the number of boxes and items, or m times n. This multiplication trick simplifies the work immensely. In our exercise, when we faced \( (x^{3/2})^{3/2} \), we utilized this rule to find that it is equal to \( x^{3/2 \cdot 3/2} = x^{9/4} \) showing us how the rule simplifies handling such expressions.

Function Composition

Function composition is a bit like putting two dance moves together to see a combined effect. In math, composing two functions is like applying one function to the result of another. Formally, if we have two functions f and g, the composition is denoted by \( f(g(x)) \) or \( g(f(x)) \), depending on the order of application. This creates a new function altogether.

It's essential to follow the order carefully, as flipping the functions might lead to a different result—just like how in a dance routine, reversing the order of moves creates a different performance. In our example from the textbook, we first applied g to f, and then f to g, to see if reversing the functions can bring us back to the original input, which is a critical step in verifying whether two functions are inverses of each other.