Question

(a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d) Based on the graph, find the range. $$f(x)=\left\\{\begin{array}{ll}-2 x+3 & \text { if } x<1 \\\3 x-2 & \text { if } x \geq 1\end{array}\right.$$

Short answer

Domain: All real numbers. Y-intercept: (0,3). No x-intercepts. Range: All real numbers.

Step-by-step solution

Finding the Domain

For the function: \[ f(x)=\left\{\begin{array}{ll}-2 x+3 & \text { if } x<1 \3 x-2 & \text { if } x \geq 1\end{array}\right. \] The domain consists of all possible values of \( x \). The function is defined for all real numbers \( x \). Hence, the domain is all real numbers, denoted as \( (-\infty, \infty) \).

Locate Intercepts

To find the intercepts:- **Finding the y-intercept:** Set \( x = 0 \) in the appropriate piece of the function. - \( 0 < 1 \), so use \( f(x) = -2x + 3 \). - \( f(0) = -2(0) + 3 = 3 \). - Thus, the y-intercept is \((0, 3)\).- **Finding the x-intercepts:** Set \( f(x) = 0 \) and solve for \( x \). - For \( x < 1 \), set \( -2x + 3 = 0 \): \( -2x + 3 = 0 \) \( -2x = -3 \) \( x = \frac{3}{2} \) - Since \( \frac{3}{2} \geq 1 \), this does not apply. Only the piece with \( x \geq 1 \). - For \( x \ge 1 \), set \( 3x - 2 = 0 \): \( 3x - 2 = 0 \) \( 3x = 2 \) \( x = \frac{2}{3} \). - Since \( \frac{2}{3} < 1 \), this does not apply. Conclusion: No x-intercepts.

Graph the Function

To draw the function:1. For \( x < 1 \), use the equation \( y = -2x + 3 \) which is a straight line.2. For \( x \geq 1 \), use the equation \( y = 3x - 2 \) which is another straight line.3. Plot the piecewise segments on the same axes: - The part \( y = -2x + 3 \) should be graphed only at points where \( x < 1 \). - The part \( y = 3x - 2 \) should be graphed only at points where \( x \geq 1 \). - Mark and connect the points appropriately.

Determine the Range

Inspecting the graph will allow identifying the range:- As \( x < 1 \), the line \( y = -2x + 3 \) descends from \( \infty \) and approaches 1.- As \( x \geq 1 \), the line \( y = 3x - 2 \) rises upwards starting from 1.- Thus, the range is all real numbers, denoted as \( (-\infty, \infty) \).

Key concepts

Domain of Piecewise Functions

The domain of a function is the set of all possible input values (typically denoted as \( x \)) that the function can accept. For our piecewise function, \[ f(x) = \begin{cases} -2x + 3 & \text{if} \; x < 1 \ 3x - 2 & \text{if} \; x \geq 1 \end{cases} \]we need to check both conditions separately.
This function is defined for all values of \( x \). The first part, \( -2x + 3 \), is valid when \( x \) is less than 1. The second part, \( 3x - 2 \), is valid when \( x \) is 1 or greater. Consequently, combining these intervals, the domain encompasses all real numbers. Therefore, we write the domain as:
\( (-\infty, \infty) \).
Understanding the domain is crucial since it tells us where the function exists and where it doesn't.

Intercepts of Piecewise Functions

Intercepts are the points where the graph of the function crosses the axes:

• The y-intercept is where the graph touches the y-axis (\( x = 0 \)).
• The x-intercepts are points where the graph crosses the x-axis (\( f(x) = 0 \)).

For the y-intercept, we use the part of the function for \( x < 1 \) since \( 0 < 1 \):
\[ f(0) = -2(0) + 3 = 3 \] Thus, the y-intercept is at (0, 3).
For the x-intercepts, set \( f(x) = 0 \) and solve for \( x \).

• For \( x < 1 \):
  \[ -2x + 3 = 0 \implies -2x = -3 \implies x = \frac{3}{2} \] Because \( \frac{3}{2} ot< 1 \), this intercept does not exist.
• For \( x \geq 1 \):
  \[ 3x - 2 = 0 \implies 3x = 2 \implies x = \frac{2}{3} \] Since \( \frac{2}{3} ot\ge 1 \), this intercept does not apply as well.

This function has:

• No x-intercepts.
• One y-intercept at (0, 3).

Graphing Piecewise Functions

Graphing piecewise functions involves drawing each segment within its specified range.

• First, let's graph \( y = -2x + 3 \) for \( x < 1 \). This is a straight line. Mark several points where \( x < 1 \) and connect them.
• Second, graph \( y = 3x - 2 \) for \( x \geq 1 \). This is also a straight line. Plot points starting from \( x = 1 \) and connect them.

To ensure clarity:

• For \( x < 1 \), you'll see a line descending linearly.
• For \( x \geq 1 \), there will be an ascending line starting from that point.

Make sure you don't graph segments outside their specific ranges. These plots will likely meet at the boundary point (1, 1) unless there's a jump at that point. For this function, there is no jump at \( x = 1 \).

Range of Piecewise Functions

The range of a function is the set of all possible output values (typically denoted as \( y \)). To find the range, inspect the graph:

• For \( x < 1 \), the line \( y = -2x + 3 \) starts from \( (0, 3) \) and descends steeply to negative infinity as x moves towards negative infinity.
• For \( x \geq 1 \), the line \( y = 3x - 2 \) rises rapidly starting from \( (1, 1) \).

Together, these segments cover all real values of \( y \), since one part decreases without bound, and the other increases without bound. Hence, the range of the given function is:
\((-\infty, \infty)\). Understanding the range helps in determining the possible outputs or values the function can produce.