Question

a) If \(f(x)=5^{x},\) state the equation of the inverse, \(f^{-1}(x)\) b) Sketch the graph of \(f(x)\) and its inverse. Identify the following characteristics of the inverse graph: \cdot the domain and range \cdot the \(x\) -intercept, if it exists \cdot the \(y\) -intercept, if it exists \cdot the equations of any asymptotes

Short answer

The inverse function is \( f^{-1}(x) = \frac{\ln(x)}{\ln(5)} \). The domain is \((0, \infty)\), the range is all real numbers, the \(x\)-intercept is \( (1, 0)\), no \(y\)-intercept, and there is a vertical asymptote at \( x = 0 \).

Step-by-step solution

Find the inverse function

To find the inverse function, start with the equation of the original function, which is given as \( f(x) = 5^x \). Swap \(x\) and \(f^{-1}(x)\) to set up the equation: \( x = 5^{f^{-1}(x)} \). Next, solve for \(f^{-1}(x)\). Take the natural logarithm of both sides to get \( \ln(x) = \ln(5^{f^{-1}(x)}) \). Using the logarithm power rule, this becomes \( \ln(x) = f^{-1}(x) \ln(5) \). Finally, solve for \(f^{-1}(x)\) by dividing both sides by \( \ln(5) \): \( f^{-1}(x) = \frac{\ln(x)}{\ln(5)} \).

Sketch the graph

To sketch the graphs of \(f(x) = 5^x\) and \(f^{-1}(x) = \frac{\ln(x)}{\ln(5)}\): - Plot \(f(x)\) by choosing some values of \(x\) and calculating corresponding \(y\) values. For instance, if \(x = -1\), \(f(-1) = 5^{-1} = 0.2\); if \(x = 0\), \(f(0) = 1\); and if \(x = 1\), \(f(1) = 5\). - Plot \(f^{-1}(x)\) using points where the coordinates are swapped from the original function. For example, if \((x,y) = (-1, 0.2)\) for \(f(x)\), then \( (y,x) = (0.2, -1)\) is a point on \(f^{-1}(x)\).

Identify characteristics of the inverse graph

To identify the domain and range of \(f^{-1}(x)\): - The domain of \(f(x)\) is all real numbers, so the range of \(f^{-1}(x)\) will be all real numbers. - The range of \(f(x)\) is \((0, \infty)\), so the domain of \(f^{-1}(x)\) will be \((0, \infty)\). Identify intercepts: - The \(x\)-intercept of \(f^{-1}(x)\) occurs when \( y = 0 \). Setting \( 0 = \frac{\ln(x)}{\ln(5)} \), solve \( \ln(x) = 0 \), which gives \( x = 1 \). So the \(x\)-intercept is \( (1, 0) \). - The \(y\)-intercept does not exist as the domain \((0, \infty)\) does not include \( x = 0 \). Identify asymptotes: - Since \(f(x) = 5^x\) has a horizontal asymptote at \( y = 0 \), \(f^{-1}(x)\) has a vertical asymptote at \( x = 0 \).

Key concepts

Understanding Exponential Functions

An exponential function is a mathematical function of the form \(f(x) = a^x\), where \(a\) is a positive real number, and \(x\) is any real number. Exponential functions have a constant base raised to a variable exponent.
They exhibit rapid growth or decay:

• For \(a > 1\), the function grows rapidly as \(x\) increases.
• For \(0 < a < 1\), the function decays rapidly as \(x\) increases.

In this exercise, the given function is \(f(x)=5^x\). Here, 5 is the base and it is greater than 1, indicating exponential growth. These functions are useful to model phenomena such as population growth, radioactive decay, and investment growth.
To find the inverse of an exponential function, we need to solve for the input variable, \(x\), in terms of the output variable, \(y\). This leads us to logarithmic functions, which we will discuss next.

Introduction to Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. For an exponential function \(f(x) = a^x\), the inverse function \(f^{-1}(x)\) is given by \(f^{-1}(x) = \frac{\text{ln}(x)}{\text{ln}(a)}\).
Here’s why this works:

• Start with the equation \(y = a^x\).
• To find the inverse, swap \(x\) and \(y\): \(x = a^y\).
• To solve for \(y\), use natural logarithms on both sides: \(\text{ln}(x) = \text{ln}(a^y)\).
• Apply the power rule of logarithms: \(\text{ln}(x) = y \text{ln}(a)\).
• Finally, isolate \(y\): \(y = \frac{\text{ln}(x)}{\text{ln}(a)}\).

In this exercise, the inverse function of \(f(x) = 5^x\) is \(f^{-1}(x) = \frac{\text{ln}(x)}{\text{ln}(5)}\).
Logarithmic functions help us solve equations involving exponential growth or decay, and are crucial for understanding the relationship between the variables in various scientific and financial contexts.

Graph Sketching Techniques

Graphing functions and their inverses provides valuable insights into their behavior.
Let’s break down the steps for sketching these graphs:

• Plotting the Exponential Function \(f(x) = 5^x\): Choose a few values for \(x\) and calculate corresponding \(y\) values. For example:
  
  • When \(x = -1\), \(f(-1) = 5^{-1} = 0.2\).
  • When \(x = 0\), \(f(0) = 1\).
  • When \(x = 1\), \(f(1) = 5\).
• Plotting the Inverse Function \(f^{-1}(x) = \frac{\text{ln}(x)}{\text{ln}(5)}\): Use the points from the original function but swap the coordinates. For example:
  
  • From \((x, y) = (-1, 0.2)\) on \(f(x)\), get \((y, x) = (0.2, -1)\) on \(f^{-1}(x)\).

Once the points are plotted:

• Domain and Range: For \(f(x)\), domain is all real numbers, and range is \((0, \rightarrow)\). For \(f^{-1}(x)\), domain is \((0, \rightarrow)\), and range is all real numbers.
• Intercepts: \(f^{-1}(x)\) has an \(x\)-intercept at (1, 0); no \(y\)-intercept as the function isn't defined at \(x = 0\).
• Asymptotes: \(f(x)\) has a horizontal asymptote at \(y = 0\); \(f^{-1}(x)\) has a vertical asymptote at \(x = 0\).

Graphing these functions helps visualize their properties and the fundamental connection between an exponential function and its logarithmic inverse.