Problem 19 - Algebra 2

Question

Find the inverse of each function. Then graph the function and its inverse. $$ g(x)=x+4 $$

Step-by-Step Solution

Answer

The inverse of the function \( g(x) = x + 4 \) is \( g^{-1}(x) = x - 4 \). Graph lines \( y = x + 4 \) and \( y = x - 4 \).

Step 1: Define the Equation for the Inverse

To find the inverse of the function \( g(x) = x + 4 \), we start by replacing \( g(x) \) with \( y \). This gives us the equation \( y = x + 4 \).

Step 2: Swap the Variables

Next, interchange the roles of \( x \) and \( y \). So, replace \( y \) with \( x \) and \( x \) with \( y \), giving us the equation \( x = y + 4 \).

Step 3: Solve for y

Now, we need to solve for \( y \). Start by isolating \( y \) on one side of the equation: \[x = y + 4\]Subtract 4 from both sides to get:\[y = x - 4\]

Step 4: Write the Inverse Function

The equation \( y = x - 4 \) represents the inverse function. Therefore, the inverse function of \( g(x) \) is \( g^{-1}(x) = x - 4 \).

Step 5: Graph the Function and Its Inverse

Graph the original function \( g(x) = x + 4 \) as a straight line with a slope of 1 and a y-intercept at 4. Then graph the inverse function \( g^{-1}(x) = x - 4 \) as another straight line with a slope of 1 and a y-intercept at -4. The intersection of these two lines with the line \( y = x \) should confirm they are inverses.

Key Concepts

Linear Functions

Linear functions are the simplest type of mathematical functions. They have the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This creates a straight line when graphed, which is why they are called linear. These functions are often used because they model relationships with constant rates of change effectively.

For example, in the function \( g(x) = x + 4 \), the slope \( (m) \) is 1. This means for each unit increase in \( x \), \( g(x) \) increases by 1. The y-intercept \( (b) \) is 4, which indicates that when \( x = 0 \), the function's output is 4.

• Understanding the slope: It tells us how much \( y \) changes for a one-unit change in \( x \).
• The y-intercept: It shows where the line crosses the y-axis.
• Linear functions depict direct relationships, meaning changes in one variable lead to proportional changes in another.

Graphing Functions

Graphing functions involves plotting points that satisfy the function on a coordinate plane. A linear function like \( g(x) = x + 4 \) is represented by a straight line because it has a constant slope. To graph it, identify key components like the y-intercept and slope and use these to plot the line.

1. **Starting Point:** Begin graphing at the y-intercept point. For \( g(x) = x + 4 \), start at \( (0, 4) \).
2. **Apply the Slope:** From the y-intercept, use the slope to find another point. Because the slope is 1, move 1 unit up for every unit you move to the right.
3. **Draw the Line:** Connect these points to form the line

The inverse function \( g^{-1}(x) = x - 4 \) also forms a line, beggining at \( (0, -4) \). Both lines will be symmetrical around the line \( y = x \). When graphing, always double-check that the points reflect the function accurately by ensuring consistency with the its slope and intercept.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations and expressions, aimed at finding specific values or simplifying equations. It is a key skill used to find inverse functions.

To find an inverse function, you swap the dependent variable (usually \( y \)) and the independent variable (usually \( x \)). This switches their roles in the equation. Next, solve for the new dependent variable to get the inverse function.

• **Step 1:** Start with the function, say, \( y = x + 4 \).
• **Step 2:** Swap \( x \) and \( y \) to get \( x = y + 4 \).
• **Step 3:** Isolate \( y \) by subtracting 4 from both sides, resulting in \( y = x - 4 \).

This simple manipulation reveals the inverse: \( g^{-1}(x) = x - 4 \). Recognizing these steps lets you maneuver through equations with confidence, uncovering relationships and simplifying solutions.