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. (a) The domain of the function \( f \) is (Type your answer in interval notation.)

The domain of the function \( f \) is all real numbers, as there are no restrictions on the values \( x \) can take. Therefore, in interval notation, the domain is \( (-\infty, \infty) \). Now, let's consider intercepts and other features! To find the y-intercept, set \( x = 0 \) in the first case of the function: \( f(0) = -2(0) + 3 = 3 \), giving the y-intercept \((0, 3)\). For the x-intercept, set \( f(x) = 0 \): 1. From \( -2x + 3 = 0 \) when \( x < 1 \), solve for \( x = \frac{3}{2} \), but \( \frac{3}{2} \geq 1 \), so no x-intercept from this case. 2. From \( 2x - 1 = 0 \) when \( x \geq 1 \), solve \( x = \frac{1}{2} \), but this \( x \) isn't in the domain of that case. Thus, the only intercept is at \( (0, 3) \). For the graph, plot the two segments: the first line, \( f(x) = -2x + 3 \) for \( x < 1 \) (which crosses the y-axis at 3 and has a slope of -2), and the second, \( f(x) = 2x - 1 \) for \( x \geq 1 \) (crossing the y-axis at -1). The range of the function from the graph is \( (-\infty, 3] \) from the first branch and \( [-1, \infty) \) from the second branch, but since the first branch hits a maximum at \( y = 3 \), the overall range is \( (-\infty, \infty) \).