Problem 25 - Algebra A Combined Function

Question

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=2 x-3 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \frac{x + 3}{2} \).

Step 1: Understand the function

The function given is \( f(x) = 2x - 3 \). This is a linear function and it is one-to-one, meaning it has a unique inverse. The goal is to find this inverse function.

Step 2: Set up the equation for inversion

To find the inverse of the function \( f(x) = 2x - 3 \), we begin by replacing \( f(x) \) with \( y \). This gives us the equation \( y = 2x - 3 \).

Step 3: Solve for x

To find the inverse, solve the equation for \( x \) in terms of \( y \). Start by adding 3 to both sides: \( y + 3 = 2x \). Then, divide both sides by 2: \( x = \frac{y + 3}{2} \).

Step 4: Express the inverse function

The expression \( x = \frac{y + 3}{2} \) is our inverse function when expressed as \( x(y) \). To express it in standard form as \( f^{-1}(x) \), swap the \( x \) and \( y \), giving us the inverse function: \( f^{-1}(x) = \frac{x + 3}{2} \).

Step 5: Graphing the functions

To graph both \( f(x) \) and \( f^{-1}(x) \), plot the function \( f(x) = 2x - 3 \) as a straight line. The slope is 2 and the y-intercept is -3. For the inverse \( f^{-1}(x) = \frac{x + 3}{2} \), the slope is 0.5 and the y-intercept is 1.5. Remember that both graphs should be mirror images across the line \( y = x \).

Key Concepts

One-to-One Function

A one-to-one function is a special type of function that has a unique output for each unique input. In simpler terms, every x-value leads to one and only one y-value. Similarly, every y-value comes from a unique x-value. This uniqueness is crucial for a function to have an inverse.

If a function is not one-to-one, its inverse wouldn't truly exist. This is because an inverse function requires reversing the process: taking y-values back to x-values uniquely. If two different x-values produced the same y-value, you wouldn't know which to choose when going backward.

• Does not have repeating y-values for different x-values.
• Passes the horizontal line test: Any horizontal line should intersect the graph of the function at most once.
• Has an inverse function that is also a one-to-one function.

Linear Function

A linear function is a function whose graph is a straight line. The general formula for a linear function is:

\[ f(x) = mx + b \]where:

• \( m \) represents the slope of the line, showing how steep it is.
• \( b \) is the y-intercept, where the line crosses the y-axis.

Linear functions are incredibly useful because they are simple and predictable. This predictability makes them easy to understand and graph.

Linear functions are always one-to-one unless the slope is zero (creating a horizontal line). This means they generally have inverses, like the function in the exercise, \( f(x) = 2x - 3 \). Here, the slope \( m \) is 2, indicating a rising line as you move from left to right.

Graphing Functions

Graphing functions is about representing functions visually on a coordinate plane. This helps us understand their behavior, like where they increase, decrease, or meet other lines.

For the function \( f(x) = 2x - 3 \):

• Identify the slope (2) and y-intercept (-3).
• Start by plotting the y-intercept: a point at (0, -3) on the y-axis.
• From this point, use the slope to find another point: go up 2 units and one unit to the right.

Draw a straight line through these points, extending it across the graph. For the inverse function \( f^{-1}(x) = \frac{x + 3}{2} \):

• The slope is 0.5, and the y-intercept is 1.5.
• Plot the point (0, 1.5) and use the slope to find another: up 0.5 units and one unit right.

When graphed together, the original and inverse functions are reflections of each other across the line \( y = x \). This line can also be drawn to visualize symmetry.

Slope and Intercept

The slope and intercept are key in understanding how a linear function behaves.

The slope is a measure of steepness and direction:

• If the slope is positive, the function rises as it moves from left to right.
• If the slope is negative, the function falls.
• A larger absolute value means a steeper line, smaller means flatter.

The y-intercept is where the function crosses the y-axis and is essential for graphing:

• It tells you the starting point of the function when x equals zero.
• For the inverse function in our case, the y-intercept is 1.5, different from the original function's -3.

Understanding these concepts is crucial for graphing effectively and predicting future points on the line. The inverse function swapping x and y changes the slope and intercept considerably, but the relationship remains linear.