Math Problem Statement
Sketch the graph of the function h(x), where h(x) is defined piecewise as:
h(x) =
x^3 - 1 if x < 1,
0 if 1 <= x <= 2,
1/x if x > 2.
Identify its domain and range.
Solution
The problem in the image is asking for the graph of the piecewise function defined as follows:
\begin{cases} x^3 - 1 & \text{if } x < 1 \\ 0 & \text{if } 1 \leq x \leq 2 \\ \frac{1}{x} & \text{if } x > 2 \end{cases}$$ We also need to identify the domain and range of the function. ### Steps to approach this: #### 1. **Graphing the function**: - For $$x < 1$$, the function is $$h(x) = x^3 - 1$$, which is a cubic function shifted down by 1. It will smoothly decrease for negative $$x$$ and approach $$-1$$ as $$x \to 1^-$$. - For $$1 \leq x \leq 2$$, the function is a constant: $$h(x) = 0$$, so this will be a flat horizontal line between $$x = 1$$ and $$x = 2$$, inclusive. - For $$x > 2$$, the function is $$h(x) = \frac{1}{x}$$, which is a hyperbola. As $$x \to \infty$$, $$h(x) \to 0$$ and as $$x \to 2^+$$, $$h(x)$$ will approach $$\frac{1}{2}$$. #### 2. **Domain of the function**: The domain is the set of all $$x$$-values where the function is defined. There are no restrictions on the value of $$x$$, so the domain is: $$\text{Domain: } (-\infty, \infty)$$ #### 3. **Range of the function**: The range is the set of all possible output values of $$h(x)$$. - For $$x < 1$$, $$x^3 - 1$$ can take values from $$-\infty$$ to $$0$$ (but not including $$0$$ because $$h(1) = 0$$). - For $$1 \leq x \leq 2$$, $$h(x) = 0$$, so $$0$$ is included. - For $$x > 2$$, $$\frac{1}{x}$$ approaches 0 from positive values, so $$h(x)$$ will take values in $$(0, \frac{1}{2}]$$. Thus, the range is: $$\text{Range: } (-\infty, 0] \cup (0, \frac{1}{2}]$$ Let me know if you'd like me to sketch the graph as well! Would you like any further clarification or details? ### Related Questions: 1. What is the behavior of the cubic function $$x^3 - 1$$ as $$x \to -\infty$$? 2. How would the graph of the function change if the condition $$1 \leq x \leq 2$$ was altered to $$1 < x < 2$$? 3. What happens to the function $$\frac{1}{x}$$ as $$x \to 2^+$$ and as $$x \to \infty$$? 4. Could you find any discontinuities in the function $$h(x)$$? If so, where? 5. How would you modify the function to ensure it is continuous over all $$x$$? ### Tip: When sketching piecewise functions, always consider the limits at the breakpoints to determine if there are any jumps or discontinuities.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Cubic functions
Constant functions
Hyperbolic functions
Formulas
h(x) = x^3 - 1 for x < 1
h(x) = 0 for 1 <= x <= 2
h(x) = 1/x for x > 2
Theorems
Domain and range of piecewise functions
Graphing piecewise functions
Suitable Grade Level
Grades 10-12