Problem 120 - Algebra and Trigonometry Real Mathematics, Real People

Question

If the inverse function of \(f\) exists, and the graph of \(f\) has a \(y\)-intercept, then the \(y\)-intercept of \(f\) is an \(x\)-intercept of \(f^{-1}.\)

Step-by-Step Solution

Answer

The statement 'If the inverse function of \(f\) exists, and the graph of \(f\) has a \(y\)-intercept, then the \(y\)-intercept of \(f\) is an \(x\)-intercept of \(f^{-1}\)' is indeed correct.

Step 1: Identify the operation on a function

When a function is inversed, the x and y are interchanged. If \(f(x) = y\), then \(f^{-1}(y) = x\).

Step 2: Apply the operation to the y-intercept

Consider the \(y\)-intercept of \(f\), which we can denote as \((0, b)\) for some \(b\). Here, \(0\) is the \(x\)-coordinate and \(b\) is the \(y\)-coordinate.

Step 3: Find the x-intercept of \(f^{-1}\)

Now let's find the corresponding point on \(f^{-1}\). Following the principle of inverse functions, we swap the \(x\)- and \(y\)-coordinates. So the point \((0, b)\) on \(f\) translates to \((b, 0)\) on \(f^{-1}\). Here \(b\) is the \(x\)-coordinate and \(0\) is the \(y\)-coordinate on \(f^{-1}\). By definition, a point with a \(y\)-coordinate of 0 is an \(x\)-intercept.

Key Concepts

Y-Intercept

Understanding the y-intercept of a function is fundamental in graphing and analyzing linear functions. The y-intercept is the point where the graph of a function crosses the y-axis. In other words, it's the value of y when the input x equals zero. To find this, you can look at the function's equation and determine the value of y when you set the x-coordinate to zero. For example, in the linear function represented as y = mx + b, b is the y-intercept and it is the point (0, b) on the graph. This will always be the starting point on the y-axis when drawing the graph of such a function. For non-linear functions, the process might require setting x to zero and solving for y, or it might be evident from the function's graph.

X-Intercept

The x-intercept is often the next step after finding the y-intercept when graphing a function. It is the location on the graph where the curve crosses the x-axis, implying that the value of y is zero. For any function, you can calculate the x-intercept(s) by setting the output y to zero and solving for x. In more complex cases where functions are not linear, there can be multiple x-intercepts, or none at all, depending on the nature of the function. Understanding both x- and y-intercepts is crucial because they provide the starting points for sketching the behavior of the graph across the coordinate plane.

Function Operations

Function operations include various ways functions can be combined or transformed. Adding, subtracting, multiplying, dividing, and composing functions are examples of such operations. In the context of inverse functions, one important operation is the inversion, where we switch the roles of the inputs and outputs. If you have a function f(x), its inverse f^{-1}(x) will give you back the original input when applied to the output of f. This concept of inversion is essential when you want to undo the action of a function, just like hitting the 'undo' button on a computer command.

Graph of a Function

The graph of a function serves as a visual representation of all the input-output pairs that make up the function. It is the set of all points (x, f(x)) in the coordinate plane where each point corresponds to one input-output pair. Understanding the graph is crucial for visual learners as it allows them to see the behavior of a function, including its increasing or decreasing intervals, symmetry, asymptotes, and intercepts. When studying the graph of the inverse function, it is excellent to note that the graph of f^{-1} will be the reflection of the graph of f across the line y = x, providing a mirror image depicting the switched roles of x and y.