Problem 92 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of \(g(x) = log_{2}(x)\) is obtained from the graph of \(f(x) = 2^{x}\) by reflecting it over the line y = x, thus the asymptote is also transformed from y = 0 in the original function to x = 0 in the inverse function.

Step 1: Understand the inverse relationship

Analyze the two given functions. The function \(f(x) = 2^{x}\) is an exponential function and \(g(x) = log_{2}(x)\) is its inverse. Therefore, the graph of \(g(x)\) can be obtained by reflecting the graph of \(f(x)\) over the line y = x.

Step 2: Consider the asymptote

The function \(f(x) = 2^{x}\) has an asymptote at y = 0. This line will be the vertical asymptote for the inverse function \(g(x) = log_{2}(x)\) due to their inverse relationship.

Step 3: Reflect key points

Find the coordinates of the key points of \(f(x)\) and reflect them over the line y = x to find key points for \(g(x)\). For instance, for \(f(x) = 2^{x}\), at x = 0, f(0) = 1, thus reflecting this point (0,1) gives us a key point (1,0) for \(g(x) = log_{2}(x)\). Similarly, when x = 1, f(1) = 2 can be reflected to (2,1) for \(g(x)\) graph.

Key Concepts

Inverse Functions

Understanding inverse functions is crucial when graphing logarithmic equations. An inverse function essentially reverses the effect of the original function. If we have a function such as f(x) = 2^{x}, its inverse will reverse the input and output. In simple terms, if the original function says 'I give you y when you give me x', the inverse function says 'OK, I’ll give you x when you give me y'.

For the function g(x) = log_{2}(x), it is the inverse of f(x) = 2^{x}. You can think of 2^{x} as asking 'what power do we need to raise 2 to get x?', whereas log_{2}(x) asks 'if 2 is raised to what number do we get x?'. Hence, the graph of the logarithmic function can be obtained by taking the graph of its exponential counterpart and flipping it over the line y = x. This property is not just a convenient trick; it's a fundamental aspect of how these functions relate to each other.

Exponential Functions

Exponential functions, such as f(x) = 2^{x}, show up frequently in real-world scenarios like population growth, finance, and radioactive decay. These functions have a constant base, in this case, 2, raised to a variable exponent. An important characteristic is that the rate of increase is proportional to the value of the function – they grow quickly!

When graphing f(x) = 2^{x}, you'll notice that as x gets larger, the value of f(x) increases exponentially, but as x becomes negative, f(x) approaches zero yet never reaches it. This behavior near zero hints at the presence of a horizontal asymptote in the graph, which is an important concept to grasp when looking at both exponential and logarithmic functions.

Asymptotes in Graphs

Understanding Horizontal and Vertical Asymptotes

Asymptotes are lines that a graph approaches but never actually touches or crosses. They can be horizontal, vertical, or even diagonal. For the function f(x) = 2^{x}, there is a horizontal asymptote at y = 0. This means that as x heads toward negative infinity, the graph gets closer and closer to the x-axis, but it will never touch it.

In the world of logarithms, like with the function g(x) = log_{2}(x), these asymptotes typically become vertical. This reflects the idea that there are values that the logarithmic function cannot take, specifically, negative numbers and zero. The graph approaches the y-axis, but again, will never cross it.

Reflection in the Line y=x

When working with inverse functions, the reflection over the line y = x is a visual representation of how each point on the graph of the original function corresponds to a point on the graph of the inverse function. This line, y = x, is where the coordinates are equal; it's a perfect 45-degree angle line that bisects the first and third quadrants of the coordinate plane.

To reflect a point across this line, you switch the x and y values of the point. For example, if you have a point (a, b) on the exponential function f(x), the corresponding point on the inverse function g(x) would be (b, a). The graph is essentially flipped over the line y = x, and this process is fundamental to understanding how to graph an inverse function based on its original function.