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 \(f^{-1}\). $$ f(x)=-2 \ln (x+1) $$

Short answer

The domain of \( f(x) \) is \( (-1, fty) \). The range is \( (-fty \) to \( fty) \), and the vertical asymptote is \( x = -1 \). The inverse function is \( f^{-1}(x) = e^{-x/2} - 1 \), with domain \( (-fty, fty) \) and range \( (-1, fty) \).

Step-by-step solution

Finding the domain of \( f(x) \)

The function is given as \( f(x) = -2 \, \text{ln} (x+1) \). The natural logarithm function, \( \text{ln}(x) \), is defined only for positive values of \( x \). Therefore, \( x + 1 > 0 \). Solving for \( x \) gives \( x > -1 \). Hence, the domain of \( f \) is \( (-1, \infty) \).

Graphing \( f(x) \)

Plotting the function involves understanding the behavior of \( -2 \, \text{ln}(x+1) \). As \( x \rightarrow -1^+ \), the term \( \text{ln}(x+1) \rightarrow -fty \), which means \( f(x) \rightarrow fty \). As \( x \rightarrow fty \), \( \text{ln}(x+1) \rightarrow fty \), thus \( f(x) \rightarrow -fty \). Consider transforming the graph of \( \text{ln}(x) \) by reflecting it vertically and scaling it.

Finding the range and asymptotes from the graph

From the graph, it is observed that \( f(x) \) ranges from \( -fty \) to \( +fty \). There is no horizontal asymptote, but there is a vertical asymptote at \( x = -1 \).

Finding the inverse function \( f^{-1}(x) \)

To find the inverse, solve \( y = -2 \, \text{ln}(x+1) \) for \( x \). Isolate the logarithmic term: \( \text{ln}(x+1) = -\frac{y}{2} \). Exponentiate both sides: \( x + 1 = e^{-y/2} \). Therefore, \( x = e^{-y/2} - 1 \). Thus the inverse function is \( f^{-1}(x) = e^{-x/2} - 1 \).

Finding the domain and range of \( f^{-1}(x) \)

The original function \( f(x) \) has the range \( (-fty, fty) \). Therefore, the domain of \( f^{-1}(x) \) is also \( (-fty, nfty) \). The original domain \( f(x) \) is \( (-1, fty) \), which becomes the range of \( f^{-1}(x) \). Hence, the range of \( f^{-1}(x) \) is \( (-1, fty) \).

Graphing \( f^{-1}(x) \)

To graph \( f^{-1}(x) \), plot the transformed exponential function \( e^{-x/2} - 1 \). The graph is an exponential decay shifted one unit downwards.

Key concepts

Domain of a Function

The domain of a function is the set of all possible input values (x-values) that the function can accept without causing any mathematical inconsistencies, like division by zero or taking the logarithm of a negative number. For the function \( f(x) = -2 \, \ln(x + 1) \), we need to ensure that the argument of the logarithm, \( x + 1 \), is positive because the logarithm is undefined for non-positive values. Hence, we solve for:
\[ x + 1 > 0 \]
This gives us \( x > -1 \). So, the domain of the function \( f(x) \) is \( (-1, \infty) \). Always be mindful of the argument inside the logarithm to determine the domain effectively.

Graphing Logarithmic Functions

Graphing logarithmic functions like \( f(x) = -2 \, \ln(x + 1) \) involves understanding the transformation of the basic logarithmic graph. The function \( \ln(x) \) has a basic shape and certain properties: it grows slowly, increases without bound as x increases, and has a vertical asymptote where the function is undefined. However, our function has additional transformations:

• The term \( x + 1 \) shifts the graph horizontally to the left by 1 unit.
• The negative sign in front reflects the graph vertically.
• The coefficient -2 compresses the graph vertically, making it steeper.

When plotting, remember that as x approaches \( -1 \) from the right, \( f(x) \) tends to infinity upward due to the vertical asymptote at \( x = -1 \). As x increases, the function decreases, reflecting the logarithmic decay.

The key points to plot would be when \( x = 0 \), and other simple values like \( x = 1 \) etc., and use the transformations mentioned to shape the graph.

Range of a Function

The range of a function is the set of all possible output values (y-values) it can produce. Based on the plot of \( f(x) = -2 \, \ln(x + 1) \), we observe how the function behaves:

• As x approaches \( -1 \) from the right, the output values of \( f(x) \) become infinitely large (positive side), hence reaching infinity.
• As x increases towards infinity, the value of \( f(x) \) decreases without bound, thus reaching negative infinity.

This behavior results in the range being all real numbers \( (-\infty, \infty) \). This is quite typical for logarithmic functions that are shifted and reflected.

Asymptotes

An asymptote is a line that a graph approaches but never touches. For our function \( f(x) = -2 \, \ln(x + 1) \):

• The vertical asymptote occurs where the function is undefined, which happens at \( x = -1 \). As x approaches \( -1 \), \( \ln(x + 1) \) trends towards negative infinity, but the negative value outside the logarithm trends the final function value towards positive infinity.
Future points can never cross this line, hence it's an important feature to highlight when graphing.

It's crucial to note vertical asymptotes, especially for functions involving logarithms or divisions, as they determine major characteristics of the function's graph.