Math Problem Statement
Do number 37
Do number 37
Solution
Let's analyze and solve number 37, where the piecewise function is defined as:
\begin{cases} 1 + x & \text{if } x < 0, \\ x^2 & \text{if } x \geq 0. \end{cases}$$ ### (a) Find the domain of $$f(x)$$: The domain of $$f(x)$$ includes all $$x$$ values for which the function is defined. Both pieces of the function are defined: - $$1 + x$$ for $$x < 0$$, which is valid for all negative $$x$$. - $$x^2$$ for $$x \geq 0$$, which is valid for all non-negative $$x$$. Thus, the **domain** of $$f(x)$$ is: $$\boxed{\text{All real numbers } (-\infty, \infty).}$$ --- ### (b) Locate any intercepts: #### $$x$$-intercept: To find the $$x$$-intercept, set $$f(x) = 0$$ and solve for $$x$$: 1. For $$x < 0$$, $$f(x) = 1 + x = 0$$: $$x = -1.$$ Therefore, there is an $$x$$-intercept at $$(-1, 0)$$. 2. For $$x \geq 0$$, $$f(x) = x^2 = 0$$: $$x = 0.$$ Thus, there is an $$x$$-intercept at $$(0, 0)$$. #### $$y$$-intercept: The $$y$$-intercept occurs when $$x = 0$$. From the second piece of the function ($$x \geq 0$$), $$f(0) = 0^2 = 0$$. Thus, the **intercepts** are: $$\boxed{(-1, 0) \text{ and } (0, 0).}$$ --- ### (c) Graph the function: The function can be sketched as: - For $$x < 0$$, $$f(x) = 1 + x$$, which is a linear function with a slope of 1 and y-intercept at $$(0, 1)$$. - For $$x \geq 0$$, $$f(x) = x^2$$, which is a parabola opening upwards, starting at $$(0, 0)$$. --- ### (d) Find the range of $$f(x)$$: 1. For $$x < 0$$, the linear part $$1 + x$$ decreases without bound as $$x \to -\infty$$. Thus, it covers $$(-\infty, 1)$$. 2. For $$x \geq 0$$, the quadratic part $$x^2$$ starts at $$0$$ and increases without bound as $$x \to \infty$$. Combining these: The **range** of $$f(x)$$ is: $$\boxed{(-\infty, \infty).}$$ Would you like me to graph this function or clarify further? 😊 --- ### Follow-Up Questions: 1. How would the domain change if $$f(x)$$ included square roots? 2. What would happen to the range if the quadratic part were modified to $$-x^2$$? 3. Can you verify the continuity of $$f(x)$$ at $$x = 0$$? 4. How do the slopes of the linear and quadratic parts compare? 5. Could you determine the derivative of $$f(x)$$ for each interval? **Tip:** Always check for continuity at points where the function's definition changes.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Intercepts
Graphing Functions
Formulas
f(x) = 1 + x for x < 0
f(x) = x^2 for x ≥ 0
Theorems
Continuity of piecewise functions
Quadratic function properties
Suitable Grade Level
Grades 9-12
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