Question

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane.\(f(x)=10^{x}, g(x)=\log _{10} x\)

Short answer

The graphs of the two functions intersect at points (1,1) and (10,1). The graph of f(x) = 10^x is an increasing curve with a horizontal asymptote at y = 0, while the graph of g(x) = log10x is also an increasing curve but with a vertical asymptote at x = 0.

Step-by-step solution

Graphing the Exponential Function

The graph of f(x) = 10^x has an x-intercept at x = 0, where y = 1. As x increases, f(x) rapidly increases. As x decreases, f(x) approaches 0 but never actually equalizes. Therefore, the x-axis is a horizontal asymptote. The range is therefore all positive values (y>0), and the domain is all real numbers.

Graphing the Inverse Function

f(x) = 10^x and g(x) = log10x are reflections of each other about the line y=x. Thus, the y-axis will reflect into a vertical asymptote, and the graph will increase as x increases but will never cross the x-axis (y=0). Here, the domain is all positive values (x>0), and the range is all real numbers.

Sketching the Graphs Together

When you plot both the functions on the same graph, you will notice that they intersect at the point (1,1) as f(1) = 10^1 = 10 and g(1) = log10(1) = 0 and also at the point (10,1). This is because f and g are inverse functions of each other.

Key concepts

Graphing Functions

When graphing functions such as exponential and logarithmic functions, it's important to understand their unique shapes and behaviors. Let's start with the exponential function. The graph of \(f(x) = 10^x\) is characterized by its dramatic rise as \(x\) increases. This is because exponential functions grow very rapidly; the further \(x\) moves to the right, the steeper the curve becomes.

Conversely, as \(x\) becomes more negative, \(f(x)\) approaches zero but never actually reaches it, giving it a horizontal asymptote along the x-axis. This means the range of the function includes all positive y-values (\(y > 0\)), while the domain is all real numbers. When plotting this, it's important to include key points such as (0,1), which is the y-intercept. This point acts as a crucial reference for sketching the rest of the curve.

On the same plane, if we consider the logarithmic function \(g(x) = \log_{10} x\), its behavior is quite different. The curve rises slowly, indicating that as \(x\) increases, \(g(x)\) increases but at a decreasing rate. Unlike the exponential graph, this function approaches a vertical asymptote along the y-axis (\(x = 0\)). The domain for this function is all positive \(x\) values (\(x > 0\)), while the range is all real numbers. Key points through which the logarithmic graph will pass include (1,0), as the log of 1 is always 0.

Understanding these foundational aspects of graphing functions sets the stage for accurately plotting and interpreting them.

Inverses of Functions

In mathematics, the concept of inverse functions unlocks a deeper understanding of how two functions relate. When two functions are inverses, each function effectively "undoes" the other. Let’s illustrate this with our functions \(f(x) = 10^x\) and \(g(x) = \log_{10} x\).

These two functions are inverses. If you input a value into \(f(x)\) that gives an output, putting that output back into \(g(x)\) will return the original input. Thus, their graphs are reflections over the line \(y = x\). This reflection is a hallmark feature of inverse relationships in graphs.

Visualizing these functions can be intuitive. Plotting both functions on the same graph helps us observe their symmetry about the line \(y = x\). This line acts as the mirror that demonstrates their inverse nature. Notice the intersecting points like (1,0) and (10,1), highlighting specific values input into one function can be mirrored by the other. Understanding the invigoration of each other's operations deepens your grasp of composition in functions.

Grasping inverse functions is fundamental, as it lays the groundwork for more advanced topics in algebra and calculus.

Asymptotes

Asymptotes are a crucial concept in understanding the behavior of functions as they extend toward infinity or encounter constraints in their domains. In the context of our functions, both \(f(x) = 10^x\) and \(g(x) = \log_{10} x\), asymptotes play significant roles.

For the exponential function \(f(x) = 10^x\), the x-axis (\(y = 0\)) is a horizontal asymptote. This means that as \(x\) approaches negative infinity, the output of \(f(x)\) nears zero, but it will never actually touch or cross this line. Understanding the horizontal asymptote here helps in predicting the function's long-term behavior.

On the other hand, \(g(x) = \log_{10} x\) involves a vertical asymptote along the y-axis (\(x = 0\)). This indicates that as \(x\) approaches zero from the positive side, the output of \(g(x)\) dives toward negative infinity. It's important to note that the function is not defined for values of \(x\) less than or equal to zero, reflecting this asymptote.

Knowing the position and type of asymptotes helps in sketching accurate graphs and predicting the behavior of functions at extremes or boundaries of their domains. Asymptotes illustrate the constraints under which functions operate, providing a more comprehensive picture of their nature.