Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Assume that \(f(x)=a^{x}\), where \(a>1\). Work these exercises in order. If the point \((p, q)\) is on the graph of \(f\), then the point _______ is on the graph of \(f^{-1}\).

Step-by-Step Solution

Answer

(q, p) is on the graph of \( f^{-1} \).

Step 1: Identify the Properties

We are given a point \(p, q\) on the graph of \(f(x) = a^x\), which is an exponential function with base \(a > 1\). We need to find the corresponding point on the inverse function graph.

Step 2: Define Inverse Function

The inverse of the function \(f(x) = a^x\) is \(f^{-1}(x) = \log_a(x)\). This inverse operation switches the roles of \(x\) and \(y\), so the point \( (q, p) \) will lie on the graph of \(f^{-1}\).

Step 3: Apply Point Transformation

Given the point \( (p, q) \) on the function \(f(x)\), the transformed point on the inverse function \(f^{-1}(x)\) will be \( (q, p) \). This is because the inverse function swaps the input and output coordinates of the original function.

Step 4: Confirm the Result

For a function and its inverse, a point \( (x, y) \) on the original function will correspond to a point \( (y, x) \) on the inverse because the function maps \(x\) to \(y\) and the inverse maps \(y\) back to \(x\). Hence, the point \( (q, p) \) is indeed on \(f^{-1}\).

Key Concepts

Inverse Functions

Inverse functions are an essential concept in mathematics, especially when dealing with exponential functions. Let's dive into what an inverse function is and how it works.
An inverse function essentially reverses the operation of the original function. If you have a function, say, \( f(x) = a^x \) where \( a > 1 \), the inverse function \( f^{-1}(x) \) will do the opposite of \( f(x) \). In this specific case, the inverse function is \( f^{-1}(x) = \log_a(x) \).
The concept of inverse functions is that they reverse the input-output relationship. For example, if a function maps \( x \) to \( y \), its inverse maps \( y \) back to \( x \). This is why when given a point \( (p, q) \) on a function, the point \( (q, p) \) will appear on its inverse. This means that if the exponential function \( f(x) \) maps a point \( (p, q) \), then its inverse \( f^{-1}(x) \) will map the point \( (q, p) \).
So, identifying points on inverse functions involves swapping the coordinates of the points on the original function.

Graph Transformations

Graph transformations are manipulations that can change the position, size, and shape of the graph of a function. Understanding these transformations helps visualize inverse functions as well.
Graph transformations can include translations, reflections, and scalings. Here, reflections are particularly important because when finding an inverse, the graph reflects across the line \( y = x \):

• Reflection: The inverse of a function reflects its graph over the line \( y = x \). This is because the inverse function swaps \( x \) and \( y \). For example, if you have the point \( (p, q) \) on \( f(x) \), it reflects to \( (q, p) \) on the inverse \( f^{-1}(x) \).

Translations, such as shifting the graph down or up, or to the left or right, do not directly apply to finding inverses, but they often come into play when transforming graphs of test functions.
Overall, graph transformations allow you to see the symmetry that exists between a function and its inverse, especially for exponential functions that invert into logarithms.

Properties of Logarithms

Logarithms are the backbone of inverse functions for exponential equations. They possess unique properties that make solving and understanding these equations simple and intuitive.
Some critical properties of logarithms include:

• Conversion from Exponential: Any exponential equation \( a^x = b \) can be converted to a logarithmic form \( x = \log_a(b) \).
• Logarithm of a Product: \( \log_a(xy) = \log_a(x) + \log_a(y) \). This property is useful when breaking down complex logarithmic expressions.
• Logarithm of a Power: \( \log_a(x^b) = b\log_a(x) \). Here, the power can be moved outside, which simplifies calculations significantly.
• Logarithm of 1: \( \log_a(1) = 0 \) because \( a^0 = 1 \).

Understanding these properties is vital when working with inverse functions of exponential equations like \( f(x) = a^x \). These rules help transform and solve exponential expressions, leading to easier computations with their inverses. By mastering these properties, students can predict how exponential functions behave and transform across different mathematical operations.