Problem 59 - A Graphical Approach to College Algebra

Question

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=x^{2}$$

Step-by-Step Solution

Answer

The function \(y = x^2\) is not one-to-one.

Step 1: Check if y = x² is one-to-one

A function is one-to-one (injective) if different inputs always produce different outputs. For \(y = x^2\), observe that \(f(2) = 4\) and \(f(-2) = 4\). Since \(f(2) = f(-2)\) but \(2 \neq -2\), the function is not one-to-one.

Step 2: Graphical interpretation

The horizontal line test confirms this: any horizontal line \(y = c\) where \(c > 0\) intersects the parabola \(y = x^2\) at two points \(x = \sqrt{c}\) and \(x = -\sqrt{c}\).

Step 3: Conclusion

Since \(y = x^2\) is not one-to-one on \(\mathbb{R}\), it does not have an inverse function (on its full domain). To make it invertible, one would need to restrict the domain to \(x \geq 0\) or \(x \leq 0\).

Key Concepts

One-to-One Function

A function is said to be one-to-one if every input has a unique output and no two different inputs produce the same output. This is crucial for a function to have an inverse that is also a function.
For example, if a function is represented by the equation \( f(x) = x^2 \), we can determine if it is one-to-one by checking if each \( y \)-value corresponds to only one \( x \)-value. However, in the case of \( y = x^2 \), both \( x = 2 \) and \( x = -2 \) will yield the same output, \( y = 4 \).
Hence, the quadratic function \( y = x^2 \) is not one-to-one since different inputs can produce the same output. If a function is not one-to-one, it cannot have an inverse that's also a function without restriction.

Horizontal Line Test

The horizontal line test is a visual way to determine if a function is one-to-one. By drawing horizontal lines across the graph of the function, you can check if any line intersects the graph more than once.
For a function to pass this test, each horizontal line should cut through the graph at most once. If it does, then the function is one-to-one, and it's possible to find an inverse function.
When you apply the horizontal line test to the parabola \( y = x^2 \), you'll notice that the horizontal lines will intersect the parabola at two points for almost any \( y \)-value. This means that the graph fails the horizontal line test, confirming that \( y = x^2 \) is not a one-to-one function.

Parabola

Parabolas are the U-shaped graphs of quadratic functions like \( y = x^2 \). These graphs are symmetric about a vertical line called the axis of symmetry. For \( y = x^2 \), this line is the \( y \)-axis.
Parabolas have several important properties:

• The vertex: This is the highest or lowest point on the graph. For \( y = x^2 \), the vertex is at the origin \((0,0)\).
• Opening direction: Since the coefficient of \( x^2 \) is positive in \( y = x^2 \), the parabola opens upwards.
• Symmetry: The left and right sides of the parabola are mirror images of each other across the axis of symmetry.

The shape of a parabola explains why the function \( y = x^2 \) is not one-to-one. As it opens upwards with a symmetrical curve, the same \( y \)-value can correspond to two different \( x \)-values, such as \( 2 \) and \( -2 \), making it impossible for the inverse to exist as a function without limiting the domain.