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

1. Given Function: \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.