Question

a) On the same set of axes, sketch the graph of the function \(y=5^{x},\) and then sketch the graph of the inverse of the function by reflecting its graph in the line \(y=x\) b) How do the characteristics of the graph of the inverse of the function relate to the characteristics of the graph of the original exponential function? c) Express the equation of the inverse of the exponential function in terms of \(y\) That is, write \(x=F(y)\)

Short answer

Graph the exponential function \(y=5^x\), reflect it over \(y=x\) to get the inverse, which is \(y=\log_5(x)\). The inverse swaps the domain and range with the original function.

Step-by-step solution

Title - Sketch the Graph of the Exponential Function

Draw the graph of the function \( y = 5^x \) on a set of axes. This graph is an exponential curve that grows rapidly, passing through the point (0, 1) and increasing as x becomes larger. Make sure to plot a few key points such as (-1, 1/5), (0, 1), (1, 5), and (2, 25).

Title - Reflect the Exponential Graph to Find the Inverse

To find the inverse of the function, reflect the graph of \( y = 5^x \) over the line \( y = x \). This reflection will give the graph of the inverse function. This new graph passes through the points (1, 0), (5, 1), and (25, 2), among others.

Title - Relationship of Characteristics

Notice that the inverse function is a logarithmic function. The domain of the exponential function \( y = 5^x \) is all real numbers and its range is \( y > 0 \). The inverse function, therefore, has a domain of \( x > 0 \) and a range of all real numbers. Also, as \( y = 5^x \) increases exponentionally, \( x = \log_5(y) \) increases logarithmically.

Title - Express the Inverse Function Equation

To find the equation of the inverse function, start with \( y = 5^x \) and switch \( x \) and \( y \) to get \( x = 5^y \). Now solve for \( y \) by taking the logarithm base 5 of both sides: \( y = \log_5(x) \). Thus, the inverse function is \( x = \log_5(y) \), which can be written as \( y = \log_5(x) \).

Key concepts

graphing exponential functions

Exponential functions like \( y = 5^x \) are powerful tools in mathematics. These functions have constant bases raised to variable exponents. The function \( y = 5^x \) specifically shows rapid growth as \( x \) increases. More specifically:

• At \( x = 0 \), \( y = 1 \) because any number to the power of zero is 1.
• At \( x = 1 \), \( y = 5 \).
• At \( x = 2 \), \( y = 25 \).
• At \( x = -1 \), \( y = \frac{1}{5} \).

Visualizing this can be done by plotting these points on a graph. Simply follow these steps:

• Draw the x-axis and y-axis on graph paper.
• Plot the points mentioned above.
• Connect these points smoothly to see the curve rise steeply to the right.

The graph of an exponential function will always pass through the point \( (0,1) \) because any nonzero number to the zero power equals one.

inverse functions

The inverse of a function 'undoes' what the original function does. For the exponential function \( y = 5^x \), its inverse can be found by reflecting its graph over the line \( y = x \). This means:

• Where the exponential function goes through \( (0,1) \), the inverse will pass through \( (1,0) \).
• Where the exponential passes through \( (1,5) \), the inverse will pass through \( (5,1) \).

In general, to find the inverse function, follow these steps:

• Start with \( y = 5^x \).
• Switch \( x \) and \( y \) to get \( x = 5^y \).
• Solve for \( y \) by applying logarithms. In this case, take the logarithm base 5 of both sides to get \( y = \log_5(x) \).

Therefore, the inverse of the exponential function \( y = 5^x \) is \( y = \log_5(x) \). This process helps us understand that the characteristics of exponential and logarithmic functions are closely related and inverted in their domains and ranges.

logarithmic functions

Logarithmic functions are the inverses of exponential functions. For example, the inverse of the function \( y = 5^x \) is \( y = \log_5(x) \). Here’s what you need to know about logarithmic functions:

• They pass through the point \( (1,0) \) because the log of 1 is always zero.
• They increase very slowly compared to exponential functions.
• They are undefined for non-positive values of \( x \).

If we compare the characteristics:

• The domain of the exponential function \( y = 5^x \) is all real numbers, while the range is \( y > 0 \).
• The domain of the logarithmic function \( y = \log_5(x) \) is \( x > 0 \), while its range is all real numbers.

Logarithms can be considered in various bases, but in this exercise, we focus on base 5, matching the base of our exponential function. This intrinsic relationship between exponentiation and logarithms makes them essential for various fields like engineering, science, and finance. Understanding both exponential and logarithmic functions allows us to solve a wide range of problems involving growth and decay, among other applications.