Question

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

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

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 \( (-\infty, \infty) \).

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.

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 \( (4, \infty) \).

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, \( f^{-1}(x) = 3 \frac{\ln(x - 4)}{\ln(2)} \).

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 \( (-\infty, \infty) \).

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.

Key concepts

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.