Problem 98 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the \(\operatorname{graph}\) of \(g(x)=\log _{2} x\)

Step-by-Step Solution

Answer

To graph \(g(x)=\log _{2} x\) using the graph of \(f(x)=2^{x}\), reflect the graph of \(f(x)=2^{x}\) across the line y = x. The graph of \(f(x)=2^{x}\) which is always increasing and crosses the point (0,1) and asymptotically approaches the x-axis will be transformed into the graph of \(g(x)=\log _{2} x\) which crosses the point (1,0) and asymptotically approaches the y-axis.

Step 1: Understanding the function \(f(x)=2^{x}\)

Firstly, to understand the given function, you should note that it's an exponential function with base 2. The graph of an exponential function is always increasing, and the graph of \(f(x)=2^{x}\) specifically passes through the point (0,1) and approaches the x-axis (y = 0) as x decreases.

Step 2: Finding the inverse function

An inverse function of \(f(x)=2^{x}\) is a function that 'reverses' the operation of the original function. Here, the inverse function of \(f(x)=2^{x}\) is \(g(x)=\log _{2} x\). So, the task is to find the graph of this inverse function \(g(x)\) using the graph of the original function \(f(x)\).

Step 3: Graphing the inverse function using the original function

Since \(f(x)\) and \(g(x)\) are inverse functions, their graphs are mirror images around the line y = x. Therefore, to graph \(g(x)=\log _{2} x\), you can reflect the graph of \(f(x)=2^{x}\) across the line y = x. The portion of \(f(x)=2^{x}\)'s graph in the first quadrant (where both x and y are positive) will reflect across y = x to form the part of \(g(x)=\log _{2} x\) in the first quadrant, and the same applies to the other quadrants.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. An example of such a function is given by f(x) = 2^x, where 2 is the base and x is the exponent. These functions are notable for their rapid growth or decay, depending on the exponent.

They have distinctive characteristics:

• Their graphs always pass through the point (0,1), since any non-zero number raised to the power of zero equals one.
• They either increase (for a base greater than one) or decrease (for a base between zero and one) exponentially as x moves away from zero.
• They approach, but never touch, the x-axis, which is known as an asymptote.

When graphing exponential functions, it's important to note these characteristics to predict the shape and behavior of the graph.

Inverse Functions

Inverse functions are pairs of functions that reverse each other's processes. In mathematical terms, if we have a function f(x), then its inverse, notated as f-1(x) (or sometimes g(x) when f(x) is given), will satisfy the condition where f(f-1(x)) = x and f-1(f(x)) = x.

For the function f(x) = 2^x, its inverse function is g(x) = log2x, where the logarithm base corresponds to the exponential function's base.

• An easy way to find the graph of an inverse function is by reflecting the graph of the original function over the line y = x.
• The domain of the original function becomes the range of the inverse, and vice versa.

Understanding these properties helps in graphing inverse functions accurately without much computation.

Function Transformation

Function transformation involves altering a parent function to shift, reflect, stretch, or compress its graph. This concept is crucial when you want to graph one function using another, as in the case of graphing g(x) = log2(x) using the graph of f(x) = 2^x.

Key transformations include:

• Reflection: Mirror a graph over a specific line, such as the y = x line for inverses.
• Translation: Shift a graph horizontally or vertically without altering its shape.
• Dilation: Stretch or compress a graph vertically or horizontally.

For instance, to create the graph of g(x) from f(x), you use a reflection transformation across the line y = x. This is because inverse functions are mirror images across this line, due to each input/output pair of the original function being swapped in the inverse.