Questions: Question 21 The function (f) is defined as follows. [ f(x)=leftbeginarrayll -3 x+4 text if x<1 3 x-2 text if x geq 1 endarrayright. ]

Transcript text: Question 21 The function $f$ is defined as follows. \[ f(x)=\left\{\begin{array}{ll} -3 x+4 & \text { if } x<1 \\ 3 x-2 & \text { if } x \geq 1 \end{array}\right. \]

Solution

Solution Steps

To find the range of the piecewise function $ f(x) $, we need to analyze each piece separately. For $ x < 1 $, the function is linear with a negative slope, and for $ x \geq 1 $, it is linear with a positive slope. We will evaluate the function at the boundary point $ x = 1 $ to ensure continuity and determine the range of each piece. The range will be the union of the ranges of these two linear functions.

Step 1: Analyze the Piecewise Function

The function $ f(x) $ is defined as: \[ f(x) = \begin{cases} -3x + 4 & \text{if } x < 1 \\ 3x - 2 & \text{if } x \geq 1 \end{cases} \] We need to determine the range of this function by analyzing each piece separately.

Step 2: Evaluate the Function at the Boundary

Evaluate the function at the boundary point $ x = 1 $ to check for continuity and determine the range at this point:

For $ x < 1 $, the function approaches $ f(1^-) = -3(1) + 4 = 1 $.
For $ x \geq 1 $, the function is $ f(1) = 3(1) - 2 = 1 $.

Step 3: Determine the Range for Each Piece

For $ x < 1 $, the function $ f(x) = -3x + 4 $ is a linear function with a negative slope. As $ x $ approaches 1 from the left, $ f(x) $ approaches 1. As $ x $ decreases, $ f(x) $ increases without bound. Thus, the range for $ x < 1 $ is $ (-\infty, 1) $.
For $ x \geq 1 $, the function $ f(x) = 3x - 2 $ is a linear function with a positive slope. As $ x $ increases, $ f(x) $ increases without bound. Thus, the range for $ x \geq 1 $ is $ [1, \infty) $.

Step 4: Combine the Ranges

The overall range of the function $ f(x) $ is the union of the ranges of the two pieces: \[ (-\infty, 1) \cup [1, \infty) = (-\infty, \infty) \]