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Chapter 6: Problem 87

Use the given function \(f\) (a) Find the domain of \(f\). (b) Graph \(f\) (c) From the graph, determine the range and any asymptotes of \(f\) (d) Fund \(f^{-1}\), the tnverse of \(f\). (e) Find the domain and the range of \(f^{-1}\). (f) Graph \(f^{-1}\). $$ f(x)=2^{x / 3}+4 $$

Short Answer

Expert verified
For \( f(x) = 2^{x/3} + 4 \), the domain is \( (-\infty, \infty) \). The inverse function is \( f^{-1}(x) = 3 \frac{\ln(x - 4)}{\ln(2)} \), with a domain of \( (4, \infty) \).

Step by step solution

01

Find the Domain of f

The domain of a function includes all the possible values of x for which the function is defined. Since the given function is an exponential function, which is defined for all real numbers, the domain of \( f(x) = 2^{x/3} + 4 \) is all real numbers, or \( (-\infty, \infty) \).
02

Graph f

To graph \( f(x) = 2^{x/3} + 4 \), plot several key points by choosing different values of x and calculating the corresponding values of f(x). Since it is an exponential function, expect the graph to rise rapidly as x increases and approach y = 4 as x decreases. Plot these points and draw the smooth curve.
03

Determine the Range and Asymptotes from the Graph

From the graph of \( f(x) = 2^{x/3} + 4 \), we can see that the function approaches, but never reaches, the horizontal line y = 4. Thus, there is a horizontal asymptote at y = 4. The range of the function is \( (4, \infty) \).
04

Find the Inverse Function

To find the inverse function \( f^{-1}(x) \), switch x and y in the definition of the original function and solve for y: \( x = 2^{y/3} + 4 \). Subtract 4 from both sides: \( x - 4 = 2^{y/3} \). Take the natural logarithm of both sides: \( \ln(x - 4) = \ln(2^{y/3}) \). Use the power rule for logarithms: \( \ln(x - 4) = \frac{y}{3} \ln(2) \). Solve for y: \( y = 3 \frac{\ln(x - 4)}{\ln(2)} \). Thus, \( f^{-1}(x) = 3 \frac{\ln(x - 4)}{\ln(2)} \).
05

Find the Domain and Range of f^{-1}

The domain of \( f^{-1}(x) \) is the range of \( f(x) \), which is \( (4, \infty) \). The range of \( f^{-1}(x) \) is the domain of \( f(x) \), which is \( (-\infty, \infty) \).
06

Graph f^{-1}

To graph \( f^{-1}(x) = 3 \frac{\ln(x - 4)}{\ln(2)} \), plot several key points. Notice that as x approaches 4 from the right, the value of \( f^{-1}(x) \) approaches negative infinity. As x increases, the function increases slowly. Plot numerous points and draw the resulting curve.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Domain
Understanding the domain of a function is crucial as it tells us the set of all possible input values (x-values) that the function can accept. For our exponential function \( f(x) = 2^{x/3} + 4 \), the function is defined for all real numbers. This is because exponential functions can handle any real number as input. Therefore, the domain is all real numbers, written as \( (-\infty, \infty) \). Whether x is negative, positive, or zero, you can always compute \( 2^{x/3} \).
Range
The range of a function represents all possible output values (y-values) it can produce. When graphing the given exponential function \( f(x) = 2^{x/3} + 4 \), we notice a few key points:
  • The function always outputs values greater than 4.
  • It never actually reaches the value 4, but it gets very close as x becomes more negative.
Thus, the smallest y-value is slightly above 4, and it increases without bound as x increases. Therefore, the y-values or the range of the function is \( (4, \infty) \).
Asymptotes
Asymptotes are lines that the function approaches but never touches. For \( f(x) = 2^{x/3} + 4 \), we see that as x decreases, the function value approaches the horizontal line \( y = 4 \). This is called a horizontal asymptote.
  • Horizontal asymptote at \( y = 4 \).
Understanding this helps in predicting the function's behavior for extreme values of x. No matter how large or small the x-value, the function will asymptotically approach, but not touch, y = 4.
Graphing Functions
Graphing functions allows us to visualize their behavior. For \( f(x) = 2^{x/3} + 4 \), follow these steps to graph it:
  • Choose a variety of x-values (both negative and positive).
  • Calculate the corresponding f(x) values.
  • Plot these points on the graph.
Notice the exponential rise as x increases and the approach towards the horizontal line \( y = 4 \) as x decreases. For the inverse function \( f^{-1}(x) = 3 \frac{\ln(x - 4)}{\ln(2)} \), repeat the plotting process. This helps you understand how the inverse reflects over the line \( y = x \). Plot key points and observe that as x approaches 4, the function heads towards negative infinity, reflecting its domain and range well.

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