Problem 80 - Precalculus Student Solutions Manual 5th

Question

Graph the inverse of each one-to-one function.

Step-by-Step Solution

Answer

Find the inverse by swapping x and y, solve for y, then reflect the original graph across y = x.

Step 1: Identify the Function

Start by identifying the one-to-one function given in the problem. Let's call this function f(x). You need the function's equation or its graph.

Step 2: Find the Inverse Function

To find the inverse function, swap x and y in the equation of the function f(x). Solve this new equation for y to get the inverse function, denoted as f^{-1}(x).

Step 3: Plot the Original Function

Graph the original function f(x) on the coordinate plane. Use known points and their coordinates to accurately plot it.

Step 4: Reflect Across y = x

To graph the inverse function, reflect the graph of f(x) across the line y = x. This means if (a, b) is a point on f(x), then (b, a) will be a point on f^{-1}(x).

Step 5: Draw the Inverse Function

Using the reflected points, draw the inverse function on the same coordinate plane. Make sure it correctly mirrors the original function.

Step 6: Verify One-to-One

Ensure that both the original function and its inverse are one-to-one. Check if each input maps to exactly one output without any repetition.

Key Concepts

One-to-One Function

A one-to-one function is a special type of function where each input value maps to a unique output value. This means no two different input values will result in the same output value.
For example, if we have a function f(x), if f(a) = f(b), then it must be the case that a = b. This uniqueness is crucial for a function to have an inverse.
Graphically, we can test if a function is one-to-one using the Horizontal Line Test. Draw horizontal lines across the graph of the function:

• If any horizontal line intersects the graph more than once, then the function is not one-to-one.
• If every horizontal line passes through at most one point on the graph, then it is a one-to-one function.

A practical example could be the function f(x) = 2x + 3, which is clearly one-to-one because for every x, there is a distinct output value.

Function Reflection

Reflecting a function involves flipping it over a specific line on the coordinate plane. For inverse functions, this line is y = x, which runs diagonally at 45 degrees through the origin.
When we reflect a function across this line, every point (a, b) on the original function f(x) will correspond to a point (b, a) on the inverse function f^{-1}(x).
To graphically find an inverse:

• Identify key points on the original function f(x).
• Swap the x and y coordinates of these points to get the coordinates of corresponding points on the inverse function.
• Plot these new points and draw the curve that passes through them.

This reflection process helps us to understand the symmetry between a function and its inverse visually.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis).
Each point on this plane can be represented by an ordered pair (x, y), where x indicates the horizontal position and y indicates the vertical position.
This plane is essential for graphing functions and their inverses.

• To graph a function, plot points corresponding to pairs of input (x) and output (y) values, and then connect these points to form a curve.
• For an inverse function, use the points obtained by swapping x and y coordinates and plot the resulting curve.

Understanding how to use the coordinate plane is fundamental for visualizing and analyzing the relationships between functions and their inverses.