Question

Graph each function. Based on the graph, state the domain and the range, and find any intercepts. $$ f(x)=\left\\{\begin{array}{ll} \ln (-x) & \text { if } x \leq-1 \\ -\ln (-x) & \text { if }-1

Short answer

Domain: \((-\infty, 0)\)Range: \( (-\infty, +\infty)\)Intercept: x-intercept at \(x = -1\)

Step-by-step solution

Understand the Piecewise Function

The function is given as a piecewise function: \[ f(x) = \begin{cases} \ln(-x) & \text{if } x \leq -1 \ -\ln(-x) & \text{if } -1 < x < 0 \end{cases} \]This means there are two different expressions for different ranges of the input \(x\).

Graph the First Part of the Function \(f(x) = \ln(-x)\)

For the range \(x\leq-1\), the function is \(f(x)=\ln(-x)\). This function is defined for all \(x\leq-1\). To graph this, take a few points in this interval:For example:For \(x=-1\), \(f(-1)=\ln(-(-1)) = \ln(1) = 0\).For \(x=-2\), \(f(-2)=\ln(-(-2)) = \ln(2) \approx 0.693\).For \(x=-3\), \(f(-3)=\ln(-(-3)) = \ln(3) \approx 1.098\).Plot these points and draw the intercept for this portion of the function.

Graph the Second Part of the Function \(f(x) = -\ln(-x)\)

For the range \(-1 < x < 0\), the function is \(f(x) = -\ln(-x)\). This function is defined for all \( -1 < x < 0 \). To graph this, take a few points in this interval:For example:For \(x=-0.5\), \(f(-0.5) = -\ln(-(-0.5)) = -\ln(0.5) \approx 0.693\). For \(x=-0.1\), \(f(-0.1) = -\ln(-(-0.1)) = -\ln(0.1) \approx 2.302\).Plot these points and draw the curve for this part of the function.

Determine the Domain

The domain of \(f(x)\) includes all values of \(x\) for which the function is defined. From the piecewise function, we see that \(f(x)\) is defined when \(x \leq -1 \) or when \(-1 < x < 0\). Thus, the domain is: \( (-\infty, 0) \).

Determine the Range

The range of \(f(x)\) includes all values that \(f(x)\) can take. For \( x \leq -1 \), \(f(x) = \ln(-x)\) which decreases without bound as \(x\) becomes more negative, approaching \( -\infty \). As \(x\) approaches \(-1\), \(\ln(-x)\) approaches \(0\). For \(-1 < x < 0\), \(f(x) = -\ln(-x)\) takes positive values and grows unbounded as \(x\) approaches \(0\) from the left. So the range of \(f(x)\) is: \( (-\infty, +\infty) \).

Find the Intercepts

To find the intercepts:- **x-intercepts**: Find where \(f(x) = 0\). From the piecewise function, it's clear there is an x-intercept at \( x = -1 \), since \(f(-1) = 0\).- **y-intercepts**: The function is not defined for \(x = 0\), so there is no y-intercept.

Key concepts

Domain and Range

Understanding the domain and range of a function is crucial when graphing it.
The **domain** tells us all the possible values for the input variable, usually denoted as 'x'.
The **range**, on the other hand, represents all the possible values for the output variable, commonly 'f(x)'. For the given piecewise function:
\[ f(x) = \begin{cases} \ln(-x) & \text{if } x \leq -1 \ \-\ln(-x) & \text{if } -1 < x < 0 \end{cases} \]
The domain combines the intervals from each part of the piecewise function:

• From \(\ln(-x)\) when \(x \leq -1\), the domain is \(x \leq -1\)
• From \(-\ln(-x)\) when \(-1 < x < 0\), the domain is \(-1 < x < 0\)

Combining these, we get the domain:
\[ (-\infty, 0) \]
The range is determined by the actual outputs from each part of the function. As x becomes more negative, \( \ln(-x) \) decreases without bound, so that part's range is \( (-\infty, 0] \). For \( -\ln(-x) \) it grows unbounded as x approaches zero from the left, giving the range for this part as \( (0, \infty) \). Combining both, the range is:
\[ (-\infty, \infty) \]

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are denoted with the 'log' notation.
The function \( \ln(x) \) denotes the natural logarithm, which is the logarithm to the base \( e \) (where \( e \approx 2.718 \)). In general, the natural logarithm function \( \ln(x) \) is only defined for positive x.
For the given piecewise function:

• \( \ln(-x) \) is only defined when \( -x > 0 \) i.e., x must be negative.
• Similarly, \( -\ln(-x) \) is defined for the same range.

This function essentially flips the usual behavior of the natural logarithm on the y-axis.
By understanding this, we can plot the values accurately.

Graphing Functions

Graphing functions typically involves plotting points and drawing lines or curves through these points.
For a piecewise function, it's essential to separately plot each portion within its defined interval.
For the given piecewise function, follow these steps:

• Graph \( \ln(-x) \) for \( x \leq -1 \).
• Graph \( -\ln(-x) \) for \(-1 < x < 0\).

By plotting several points from each interval, you can discern the shape and direction of each segment.
For example:

• For \( f(x) = \ln(-x) \), some points are \( (-1, 0) (-2, 0.693) \) and so forth.
• For \( f(x) = -\ln(-x) \), points include \( (-0.5, 0.693) (-0.1, 2.302) \).

Connect these points smoothly to reflect the behavior accurately.

Intercepts

Intercepts are the points where the function crosses the axes.
There are two types of intercepts:
**x-intercepts and y-intercepts**. The x-intercepts are where the function value is zero.
The given piecewise function has an x-intercept at \( x = -1 \) because it satisfies:
\( f(-1) = \ln(-(-1)) = \ln(1) = 0 \).
The y-intercept would occur where \( x = 0 \), but in this case, the function is not defined for \( x = 0 \). Hence, there is no y-intercept.
Remember, finding intercepts helps understand the function’s behavior relative to the axes.