Math Problem Statement
The function f is defined as f(x) = {3 + x if x < 0, x^2 if x ≥ 0}. (a) Find the domain of the function. (b) Locate any intercepts. (c) Graph the function. (d) Based on the graph, find the range.
Solution
Let's solve the questions about the given function .
The function is defined piecewise as:
3 + x, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases}$$ ### (a) Find the domain of the function. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case: - For $$x < 0$$, the function is $$3 + x$$, which is defined for all negative $$x$$. - For $$x \geq 0$$, the function is $$x^2$$, which is defined for all non-negative $$x$$. Thus, the domain of $$f(x)$$ is all real numbers $$(-\infty, \infty)$$. ### (b) Locate any intercepts. - **x-intercepts**: These occur where $$f(x) = 0$$. - For $$x < 0$$, solve $$3 + x = 0$$ ⇒ $$x = -3$$. - For $$x \geq 0$$, solve $$x^2 = 0$$ ⇒ $$x = 0$$. - So, the x-intercepts are at $$x = -3$$ and $$x = 0$$. - **y-intercept**: This occurs where $$x = 0$$. At $$x = 0$$, $$f(0) = 0^2 = 0$$. So, the y-intercept is at $$(0, 0)$$. ### (c) Graph the function. - For $$x < 0$$, the graph is the line $$f(x) = 3 + x$$, which has a slope of 1 and a y-intercept at $$(0, 3)$$, but this part only applies for $$x < 0$$. - For $$x \geq 0$$, the graph is the parabola $$f(x) = x^2$$, starting at $$(0, 0)$$. The graph consists of two parts: 1. A line with slope 1 for $$x < 0$$. 2. A parabola starting at $$(0, 0)$$ for $$x \geq 0$$. ### (d) Based on the graph, find the range. - For $$x < 0$$, $$f(x) = 3 + x$$, so the values of $$f(x)$$ range from $$(-\infty, 3)$$. - For $$x \geq 0$$, $$f(x) = x^2$$, so the values of $$f(x)$$ range from $$[0, \infty)$$. Therefore, combining both parts, the range of the function is $$(-\infty, 3) \cup [0, \infty)$$. Would you like further details on any specific part of this solution? --- Here are 5 related questions to expand on this topic: 1. How do we determine whether a function is continuous or not? 2. What is the difference between the domain and the range of a function? 3. How do piecewise functions apply in real-world problems? 4. Can the function have a different slope in other intervals? 5. How do you calculate the derivative of a piecewise function? **Tip**: When graphing piecewise functions, always pay attention to whether the conditions include the boundary point, such as using $$\geq$$ or $$>$$ at key points.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Domain and Range
Intercepts
Graphing
Formulas
f(x) = 3 + x (for x < 0)
f(x) = x^2 (for x ≥ 0)
x-intercept: solve f(x) = 0
y-intercept: f(0)
Theorems
Definition of Domain and Range
Graphing piecewise functions
Intercepts of a function
Suitable Grade Level
Grades 9-12
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