Question

In Problems \(30-34,\) find the inverse function. $$ h(x)=2 x+4 $$

Short answer

Answer: The inverse function of \(h(x) = 2x + 4\) is \(h^{-1}(x) = \frac{x - 4}{2}\).

Step-by-step solution

Rewrite the function with y

Before we find the inverse, let's rewrite the given function, \(h(x) = 2x + 4\), in terms of \(y\): $$ y = 2x + 4 $$

Swap x and y variables

Now, we need to swap \(x\) and \(y\) in order to find the inverse function. This means, wherever we see \(x\), we replace it with \(y\), and wherever we see \(y\), we replace it with \(x\): $$ x = 2y + 4 $$

Solve for y

After swapping the variables, our goal is to solve for the new \(y\) in terms of \(x\). In order to do this, we need to isolate \(y\) on one side of the equation: 1. First, subtract 4 from both sides: $$ x - 4 = 2y $$ 2. Next, divide both sides by 2: $$ \frac{x - 4}{2} = y $$

Rewrite as the inverse function

Now that we have found the value of \(y\) in terms of \(x\), we can rewrite the function as the inverse function \(h^{-1}(x)\): $$ h^{-1}(x) = \frac{x - 4}{2} $$ So, the inverse function of \(h(x) = 2x + 4\) is: $$ h^{-1}(x) = \frac{x - 4}{2} $$

Key concepts

Function Notation

Function notation is a way to express the relationship between inputs and outputs in mathematics. It makes it easy to see how each input into a function results in an output. This is done using a letter, commonly "f," to represent the function, and "x" representing the input, like so: \( f(x) \).
This reads as "f of x," indicating that "f" is a function with "x" as its input. For example, if you have a function \( h(x) = 2x + 4 \), it means that for each value of "x," you multiply it by 2 and add 4 to get the output. **Why use function notation?**

• It clearly conveys the operation being performed on the variable.
• It helps in defining inverse functions by easily swapping variables.
• It provides a consistent method for representing functions, making equations easier to work with.

Function notation is not just a formality; it plays a critical role in organizing and solving mathematical problems, especially when dealing with complex expressions.

Solving Equations

Solving equations is like finding the treasure hidden in a chest. You know the result and need to work backward to find the original numbers that give the answer.
When we talk about solving equations, we specifically mean finding the value of the variable that makes the equation true. Here’s a step-by-step look at the general process:
1. **Identify the Equation:** Recognize what you are dealing with, such as \( x = 2y + 4 \), after swapping the variables in our inverse function problem.2. **Perform Inverse Operations:** Use addition, subtraction, multiplication, or division to isolate the variable on one side of the equation.

• For \( x = 2y + 4 \), subtract 4 from both sides to get \( x - 4 = 2y \).
• Then, divide both sides by 2 to solve for \( y \), resulting in \( y = \frac{x - 4}{2} \).

**Why Solving Equations Matters:**
Solving equations forms the bedrock of algebra as it allows us to reverse operations—like solving mysteries with clues. It helps check if our inverses or solutions are right and is an invaluable skill in everyday mathematics.

Linear Functions

Linear functions are like straight roads in math—they have constant slopes and create straight-line graphs. The equation for a linear function takes the form \( y = mx + b \), where "m" is the slope, and "b" is the y-intercept.
**Understanding the Components:**

• **Slope (\