Question 2.4.10: Let f(x)=x^2/3-3. (a) Find f(-2). (b) Sketch the graph of f.......
Let f(x)=x^{2/3}-3.
(a) Find f(-2).
(b) Sketch the graph of f.
(c) State the domain and range of f.
(d) State the intervals on which f is increasing or is decreasing.
(e) Estimate the x-intercepts of the graph to one-decimal-place accuracy.
Final Answer
(a) Shown to the right are four representations of f. All of these are valid on the TI-83/4 Plus. On some older graphing calculator models, you may get only the right-hand side of the graph in Figure 14 below. If that happens, change your representation of f.
Shown to the right are two methods of finding a function value. In the first method, we simply find the value of Y_1(-2). In the second method, we store -2 in \mathrm{X} and then find the value of \mathrm{Y}_1.
\boxed {\text { VARS }} \quad \boxed {\vartriangleright} \quad \quad \boxed {1}\quad \boxed {1}\quad \boxed {(} \quad -2 \quad \boxed {)} \quad \boxed {\text { ENTER }}
(b) Using the viewing rectangle [-15, 15] by [-10, 10] to graph Y_1 gives us a display similar to that of Figure 14. The v-shaped part of the graph of f at x=0 is called a cusp.
(c) The domain of f is \mathbb{R}, since we may input any value for x. The figure indicates that y ≥-3, so we conclude that the range of f is [-3, ∞).
(d) From the figure, we see that f is decreasing on (-∞, 0] and is increasing on [0, ∞).
(e) Using the root feature, we find that the positive x-intercept in Figure 14 is approximately 5.2. Since f is symmetric with respect to the y-axis, the negative x-intercept is about -5.2 .
