Problem 111 - A Graphical Approach to Precalculus with Limits

Question

Use a graph with the given viewing window to decide which functions are one- to-one. If a function is one-to-one, give the equation of its inverse function. Check your work by graphing the inverse function on the same coordinate axes. $$f(x)=\frac{x-5}{x+3} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

Step-by-Step Solution

Answer

The function is one-to-one. Inverse: \(f^{-1}(x) = \frac{-3x - 5}{x - 1}\).

Step 1: Understand One-to-One Functions

A function is one-to-one if each output is determined by exactly one input. Graphically, this means the function passes the Horizontal Line Test, where no horizontal line intersects the graph more than once.

Step 2: Analyze the Given Function Graphically

Graph the function \( f(x) = \frac{x-5}{x+3} \) within the viewing window of \([-6.6, 6.6] \) for the x-values and \([-4.1, 4.1] \) for the y-values. Look for sections where a horizontal line might intersect the graph multiple times.The function should appear as a hyperbola with asymptotes at \( x = -3 \) (vertical) and as \( y = 1 \) (horizontal) because as \( x \to \infty, f(x) \to 1 \). Confirm visually that no horizontal line crosses the function more than once.

Step 3: Confirm One-to-One Property

In the inspection within the viewing window, observe that no horizontal lines intersect the graph of \( f(x) \) more than once. Thus, \( f(x) = \frac{x-5}{x+3} \) is a one-to-one function.

Step 4: Derive the Inverse Function

To find the inverse, swap \( x \) and \( y \) in the function equation and solve for \( y \):\[x = \frac{y-5}{y+3}\]Cross-multiply to obtain:\[x(y+3) = (y-5)\]Expand the equation:\[xy + 3x = y - 5\]Rearrange terms to isolate \( y \):\[xy - y = -3x - 5\]Factor \( y \):\[y(x - 1) = -3x - 5\]Solve for \( y \):\[y = \frac{-3x - 5}{x - 1}\]The inverse function is \( f^{-1}(x) = \frac{-3x - 5}{x - 1} \).

Step 5: Graph the Inverse Function

Place the graph of the inverse function \( f^{-1}(x) = \frac{-3x - 5}{x - 1} \) alongside the original function on the same coordinate axes using the same viewing window \([-6.6, 6.6] \) by \([-4.1, 4.1]\). Check that both functions are symmetrical about the line \( y = x \). This confirms the correctness of the inverse.

Key Concepts

Horizontal Line Test

The horizontal line test is an essential tool to determine whether a function is "one-to-one." If a function passes this test, it means that each output value is paired with exactly one input value. Here’s how it works:
When you draw horizontal lines across the graph of the function, each line should intersect the graph at most once.
Breaking it down further:

• If a horizontal line crosses the function more than once, the function fails the test, and it is not considered one-to-one.
• If no horizontal line crosses the function more than once, the function passes the test, and it is one-to-one.

In our problem, the function \( f(x) = \frac{x-5}{x+3} \) must be examined to see if any horizontal line would touch the graph more than once.
For this specific function, the test involves analyzing its graphical behavior. With a careful review within the set viewing window, the function doesn’t intersect any horizontal line more than once, confirming it's one-to-one.

Inverse Functions

An inverse function essentially reverses the input and output of the original function. For a function to have an inverse, it needs to be one-to-one, ensuring every y-value comes from only one x-value.
Once we've established that a function is one-to-one using the horizontal line test, we can proceed to find its inverse.
Here's how to find the inverse function step-by-step:

• Swap the x and y in the equation of the original function. For the given function \(f(x) = \frac{x-5}{x+3}\), rewrite it as \(x = \frac{y-5}{y+3}\).
• Solve the resulting equation for y to get it in terms of x. Cross-multiply to get rid of the fraction: \(x(y+3) = (y-5)\).
• Simplify and rearrange to isolate y:
  1. Expand to \(xy + 3x = y - 5\).
  2. Manipulate to \(xy - y = -3x - 5\).
  3. Factor out y from the left side and solve: \(y(x - 1) = -3x - 5\).
  4. Finally, solve for y: \( y = \frac{-3x - 5}{x - 1}\).

The result is the inverse function \(f^{-1}(x) = \frac{-3x - 5}{x - 1}\).

Graphical Analysis

Graphical analysis plays a crucial role in understanding functions and their inverses.
When you graph both the original function and its inverse on the same set of axes, comparing their behaviors becomes intuitive.
Key insights from graphical analysis include:

• The original function \( f(x) \) and its inverse \( f^{-1}(x) \) should be symmetrical about the line \( y = x \).
• Plotting each function in the same viewing window—such as \([-6.6, 6.6]\) for x and \([-4.1, 4.1]\) for y—helps verify this symmetry.
• Symmetry about \( y = x \) confirms not only that the inverse was calculated correctly but also that the original function was indeed one-to-one.

In this exercise, examining the plots of \( f(x) = \frac{x-5}{x+3} \) and \( f^{-1}(x) = \frac{-3x - 5}{x - 1} \), their reflection across \( y = x \) solidifies understanding, ensuring confidence in the function’s characteristics.
Analyzing these visual cues provides deeper insights into how inverse functions interact with their counterparts on a graph.