Math Problem Statement

The function f is defined as follows. f left parenthesis x right parenthesis equalsleft brace Start 2 By 1 Matrix 1st Row 1st Column 4 plus x 2nd Row 1st Column x squared EndMatrix Start 2 By 1 Matrix 1st Row 1st Column if x less than 0 2nd Row 1st Column if x greater than or equals 0 EndMatrix

​(a) Find the domain of the function. ​(b) Locate any intercepts. ​(c) Graph the function. ​(d) Based on the​ graph, find the range. Question content area bottom Part 1 ​(a) The domain of the function f is    enter your response here. ​(Type your answer in interval​ notation.)

Solution

(a) Domain of the function

The function f(x)f(x) is defined as:

\begin{cases} 4 + x & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$$ The domain of a function consists of all possible values of $$x$$ for which the function is defined. Since both pieces of the function $$4 + x$$ and $$x^2$$ are defined for all real values of $$x$$, the domain includes all real numbers. Thus, the domain is: $$\boxed{(-\infty, \infty)}$$ ### (b) Intercepts #### x-intercept: - The x-intercept occurs where $$f(x) = 0$$. For $$x < 0$$, $$f(x) = 4 + x$$. Set $$4 + x = 0$$: $$4 + x = 0 \implies x = -4$$ For $$x \geq 0$$, $$f(x) = x^2$$. Set $$x^2 = 0$$: $$x = 0$$ Thus, the x-intercepts are $$x = -4$$ and $$x = 0$$. #### y-intercept: - The y-intercept occurs where $$x = 0$$. At $$x = 0$$, we use the second piece of the function since $$x = 0 \geq 0$$. So, $$f(0) = 0^2 = 0$$. Therefore, the y-intercept is $$(0, 0)$$. ### (c) Graph of the function The function consists of two parts: 1. For $$x < 0$$, the function is a linear function $$f(x) = 4 + x$$. 2. For $$x \geq 0$$, the function is a quadratic function $$f(x) = x^2$$. At $$x = 0$$, the two pieces of the function meet at $$(0, 0)$$, and the graph transitions smoothly from the linear piece to the quadratic piece. I will generate a graph of this function. ### (d) Range of the function Based on the graph: - For $$x < 0$$, $$f(x) = 4 + x$$, which decreases as $$x$$ decreases. The maximum value is $$f(0) = 0$$, and as $$x \to -\infty$$, $$f(x) \to -\infty$$. - For $$x \geq 0$$, $$f(x) = x^2$$, which is non-negative and increases as $$x$$ increases. The minimum value is $$f(0) = 0$$, and there is no upper bound. Thus, the range is: $$\boxed{(-\infty, \infty)}$$ --- Would you like me to graph the function for you or need more details? ### Related questions: 1. How do you find the range from the equation alone? 2. What is the significance of the domain in piecewise functions? 3. Can there be more than one y-intercept for piecewise functions? 4. How do quadratic functions affect the shape of graphs? 5. Why does the function's behavior change at $$x = 0$$? **Tip:** For piecewise functions, make sure to evaluate the function at the boundary points of each piece to ensure continuity or determine any breaks in the graph.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Domain and Range
Graphing
Intercepts

Formulas

f(x) = 4 + x for x < 0
f(x) = x^2 for x ≥ 0

Theorems

Piecewise Function Definition
Quadratic Function Properties
Linear Function Properties

Suitable Grade Level

Grades 9-11