Exercise 2 Let \( f(x)=\left\{\begin{array}{ll}x^{2}+2 x, & \text { if } x \leq-1 \\ x, & \text { if }-11\end{array} \quad\right. \) a piecewise defined function. a) What is the domain? b) Find \( f(-4), f\left(-\frac{3}{2}\right), f(-1), f(0), f(25) \)

The function \( f(x) \) is defined for all real numbers, so the domain is \( (-\infty, \infty) \). Now, let's evaluate the function at different points: a) The domain is \( \mathbb{R} \). b) - \( f(-4) = (-4)^2 + 2(-4) = 16 - 8 = 8 \) (because \( -4 \leq -1 \)) - \( f\left(-\frac{3}{2}\right) = \left(-\frac{3}{2}\right)^2 + 2\left(-\frac{3}{2}\right) = \frac{9}{4} - 3 = \frac{9}{4} - \frac{12}{4} = -\frac{3}{4} \) (because \( -\frac{3}{2} \leq -1 \)) - \( f(-1) = -1 \) (because \( -1 \) is included in the case \( x \leq -1 \)) - \( f(0) = 0 \) (because \( 0 \) falls in the case \( -1 < x \leq 1 \)) - \( f(25) = -1 \) (because \( 25 > 1 \)). To summarize: - \( f(-4) = 8 \) - \( f\left(-\frac{3}{2}\right) = -\frac{3}{4} \) - \( f(-1) = -1 \) - \( f(0) = 0 \) - \( f(25) = -1 \)