The function $$ f $$ is defined as follows. $$ f(x)=\left\{\begin{array}{ll}-2 x+3 & \text { if } x

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

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Let's analyze the function $$ f $$ defined as follows:

$$
f(x)=\left\{\begin{array}{ll}-2 x+3 & \text { if } x<1 \ 2 x-1 & \text { if } x \geq 1\end{array}\right.
$$

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

The function $$ f(x) $$ is defined for all real numbers $$ x $$. There are no restrictions on $$ x $$ in either piece of the piecewise function. Therefore, the domain of $$ f $$ is:

$$
\text{Domain: } (-\infty, \infty)
$$

### (b) Locate any intercepts.

**Finding the y-intercept:**
To find the y-intercept, we evaluate $$ f(0) $$:

Since $$ 0 < 1 $$, we use the first piece of the function:

$$
f(0) = -2(0) + 3 = 3
$$

Thus, the y-intercept is at the point $$ (0, 3) $$.

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

1. For $$ x < 1 $$:
   $$
   -2x + 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}
   $$
   However, $$ \frac{3}{2} = 1.5 $$ is not in the domain $$ x < 1 $$, so there is no x-intercept from this piece.

2. For $$ x \geq 1 $$:
   $$
   2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}
   $$
   However, $$ \frac{1}{2} = 0.5 $$ is not in the domain $$ x \geq 1 $$, so there is no x-intercept from this piece.

Thus, there are no x-intercepts.

### Summary of intercepts:
- **Y-intercept:** $$ (0, 3) $$
- **X-intercept:** None

### (c) Graph the function.

To graph the function, we need to plot the two pieces:

1. For $$ x < 1 $$: The line $$ f(x) = -2x + 3 $$ has a y-intercept at $$ (0, 3) $$ and a slope of -2. It will cross the x-axis at $$ (1.5, 0) $$, but we only consider the part of the line for $$ x < 1 $$.

2. For $$ x \geq 1 $$: The line $$ f(x) = 2x - 1 $$ has a y-intercept at $$ (0, -1) $$ (not relevant here) and a slope of 2. At $$ x = 1 $$:
   $$
   f(1) = 2(1) - 1 = 1
   $$
   So the point $$ (1, 1) $$ is included in this piece.

The graph will consist of:
- A line segment from $$ (0, 3) $$ to $$ (1, 1) $$ for $$ x < 1 $$ (open circle at $$ (1, 1) $$).
- A line starting from $$ (1, 1) $$ and going upwards for $$ x \geq 1 $$.

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

From the graph:
- The first piece $$ f(x) = -2x + 3 $$ decreases from $$ (0, 3) $$ to $$ (1, 1) $$, covering the interval $$ (1, 3] $$.
- The second piece $$ f(x) = 2x - 1 $$ starts at $$ (1, 1) $$ and increases without bound as $$ x $$ increases.

Thus, the range of the function is:

$$
\text{Range: } [1, \infty)
$$

### Final Summary:
- **Domain:** $$ (-\infty, \infty) $$
- **Y-intercept:** $$ (0, 3) $$
- **X-intercept:** None
- **Range:** $$ [1, \infty) $$
- Domain: $$ (-\infty, \infty) $$ 	- Y-intercept: $$ (0, 3) $$ 	- X-intercept: None 	- Range: $$ [1, \infty) $$

The function is defined as follows.

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

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The function is defined as follows. (a) Find the domain of the function. (b) Locate any intercepts. © Graph the function. (d) Based on the graph, find the range.