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To solve this piecewise-defined function problem, let's go through each part of the question step-by-step.

### Given Function
The function \( f(x) \) is defined as:
\[
f(x) = 
\begin{cases} 
-3x + 5 & \text{if } x < 1 \\
4x - 2 & \text{if } x \geq 1 
\end{cases}
\]

### Part (a): Domain of the Function
The domain of a function consists of all possible values of \( x \) for which \( f(x) \) is defined.

- For \( x < 1 \), \( f(x) = -3x + 5 \) is defined.
- For \( x \geq 1 \), \( f(x) = 4x - 2 \) is defined.

Since both parts of the function cover all real numbers without any restrictions, the domain of \( f \) is all real numbers. In interval notation, this is:
\[
(-\infty, \infty)
\]

### Part (b): Intercepts
To find the intercepts, we look for points where \( f(x) = 0 \) (x-intercepts) and where \( x = 0 \) (y-intercepts).

#### X-Intercept(s)
1. **For \( x < 1 \):** Set \( -3x + 5 = 0 \).
   \[
   -3x + 5 = 0 \Rightarrow x = \frac{5}{3}
   \]
   Since \( \frac{5}{3} \approx 1.67 \) (greater than 1), this solution does not fall in the interval \( x < 1 \), so there is no x-intercept for this part of the function.

2. **For \( x \geq 1 \):** Set \( 4x - 2 = 0 \).
   \[
   4x - 2 = 0 \Rightarrow x = \frac{1}{2}
   \]
   Since \( \frac{1}{2} < 1 \), this does not fall in the interval \( x \geq 1 \), so there is no x-intercept for this part of the function either.

There are no x-intercepts for \( f(x) \).

#### Y-Intercept
To find the y-intercept, set \( x = 0 \).
- For \( x = 0 \), we use the part of the function where \( x < 1 \), which is \( f(x) = -3x + 5 \).
  \[
  f(0) = -3(0) + 5 = 5
  \]
  So the y-intercept is \( (0, 5) \).

### Part (c): Graph the Function
To graph \( f(x) \), plot each piece on its respective interval.

1. **For \( x < 1 \):** \( f(x) = -3x + 5 \)
   - This is a linear function with slope \( -3 \) and y-intercept \( 5 \).
   - For \( x \to 1^- \), \( f(x) \to -3(1) + 5 = 2 \).
   - So, in this region, we have a line passing through \( (0, 5) \) and approaching \( (1, 2) \) from the left.

2. **For \( x \geq 1 \):** \( f(x) = 4x - 2 \)
   - This is a linear function with slope \( 4 \) and y-intercept \( -2 \).
   - For \( x = 1 \), \( f(x) = 4(1) - 2 = 2 \).
   - So, in this region, we have a line starting at \( (1, 2) \) and continuing with a positive slope.

The graph will show a continuous function with no gaps at \( x = 1 \).

### Part (d): Range of the Function
The range is the set of all possible values of \( f(x) \).

- As \( x \to -\infty \), \( f(x) = -3x + 5 \) will tend towards \( +\infty \).
- As \( x \to +\infty \), \( f(x) = 4x - 2 \) will also tend towards \( +\infty \).
- The minimum value of \( f(x) \) occurs at \( f(1) = 2 \).

Therefore, the range of \( f(x) \) is:
\[
[2, \infty)
\]

---

Would you like more details on any part of this solution? 

Here are some additional questions to deepen understanding:
1. How would the domain change if one of the intervals was restricted to a smaller subset?
2. How would the function's continuity be affected if there was a gap or jump at \( x = 1 \)?
3. Can you identify other functions where the range is similarly constrained by a minimum or maximum value?
4. What happens to the intercepts if the function was transformed with a vertical shift?
5. How would the graph look if the slope of each piece was reversed?

**Tip:** When working with piecewise functions, always evaluate boundary points carefully to ensure continuity or identify any potential jumps.

The function f is defined as follows:

f(x) =
-3x + 5 if x < 1
4x - 2 if x >= 1

(a) Find the domain of the function.
(b) Locate any intercepts.
(c) Graph the function.
(d) Based on the graph, find the range.

Learn how to analyze a piecewise function with two linear equations by finding its domain, intercepts, graph, and range. This example covers fundamental concepts of domain and range in functions, suitable for high school students.

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