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Let's go step by step through each part of the problem involving the piecewise function \( f(x) \):

1. **Given Function:**
   \[
   f(x) = 
   \begin{cases} 
   -2x + 3 & \text{if } x < 1 \\
   4x - 3 & \text{if } x \geq 1 
   \end{cases}
   \]

### (a) Find the domain of the function.

The domain of a function is the set of all possible input values (x-values). In this case:
- For \( x < 1 \), the function is \( -2x + 3 \), which is valid for all \( x \) less than 1.
- For \( x \geq 1 \), the function is \( 4x - 3 \), which is valid for all \( x \) greater than or equal to 1.

Thus, the domain of the function is **all real numbers**:
\[
\text{Domain: } (-\infty, \infty)
\]

### (b) Locate any intercepts.

#### i) **x-intercept:**
To find the x-intercept, we set \( f(x) = 0 \) and solve for \( x \).

- For \( x < 1 \) (using \( f(x) = -2x + 3 \)):
  \[
  -2x + 3 = 0 \implies x = \frac{3}{2} = 1.5
  \]
  But this is outside the valid range of \( x < 1 \), so there is no x-intercept in this case.

- For \( x \geq 1 \) (using \( f(x) = 4x - 3 \)):
  \[
  4x - 3 = 0 \implies x = \frac{3}{4} = 0.75
  \]
  Since \( x = 0.75 \) is less than 1, it doesn't fit the range of \( x \geq 1 \), so there is **no x-intercept** in this case.

#### ii) **y-intercept:**
To find the y-intercept, set \( x = 0 \) and find \( f(0) \).

Since \( 0 < 1 \), we use \( f(x) = -2x + 3 \):
\[
f(0) = -2(0) + 3 = 3
\]

Thus, the y-intercept is \( (0, 3) \).

### (c) Graph the function.

To graph the function:
- For \( x < 1 \), plot the line \( f(x) = -2x + 3 \), which has a y-intercept of 3 and a slope of -2.
- For \( x \geq 1 \), plot the line \( f(x) = 4x - 3 \), which starts at \( x = 1 \) and has a slope of 4.

At \( x = 1 \):
- \( f(1) = 4(1) - 3 = 1 \). So, there is a point at \( (1, 1) \).
  
The graph will show two distinct lines: one line before \( x = 1 \) and another starting from \( x = 1 \).

### (d) Based on the graph, find the range.

The range is the set of all possible output values (y-values). Based on the two parts of the function:

- For \( x < 1 \), the function \( -2x + 3 \) decreases without bound as \( x \to -\infty \), so the lower limit of the range is \( -\infty \).
- For \( x \geq 1 \), the function \( 4x - 3 \) increases without bound as \( x \to \infty \).

The range of the function is:
\[
\text{Range: } (-\infty, \infty)
\]

---

Do you need more details on any part of the solution? Here are some related questions to further explore this topic:

1. What happens to the graph of the function as \( x \to \infty \)?
2. How do you calculate the slope of a linear function?
3. What is the significance of discontinuity in a piecewise function like this one?
4. How can you check if the piecewise function is continuous at \( x = 1 \)?
5. Can there be more than one y-intercept in a piecewise function?

**Tip:** When graphing piecewise functions, always check for continuity and whether the function's values match at the boundaries between intervals.

The function f is defined as follows:
 f(x) = -2x + 3 for x < 1; 4x - 3 for 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 work with a piecewise function defined as f(x) = -2x + 3 for x < 1 and f(x) = 4x - 3 for x ≥ 1. Find the domain, intercepts, graph the function, and determine its range. Suitable for students in Grades 10-12.

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