Math Problem Statement
The function f is defined as a piecewise function: f(x) = -2x + 3 if x < 1 and f(x) = 2x - 1 if x >= 1. The tasks are to find the domain, locate intercepts, graph the function, and determine the range.
Solution
The given piecewise function is defined as:
-2x + 3 & \text{if } x < 1 \\ 2x - 1 & \text{if } x \geq 1 \end{cases}$$ We need to solve the following parts: ### (a) Find the domain of the function. The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, both pieces of the function are defined for all real numbers. The first piece $$-2x + 3$$ applies when $$x < 1$$, and the second piece $$2x - 1$$ applies when $$x \geq 1$$. Thus, the domain is all real numbers: $$\text{Domain of } f(x): (-\infty, \infty)$$ ### (b) Locate any intercepts. #### Finding the y-intercept: The y-intercept occurs when $$x = 0$$. We use the first piece of the function since $$x = 0 < 1$$: $$f(0) = -2(0) + 3 = 3$$ Thus, the y-intercept is $$(0, 3)$$. #### Finding the x-intercept(s): For the x-intercept(s), set $$f(x) = 0$$ and solve for $$x$$ in each piece. - For $$x < 1$$, $$-2x + 3 = 0$$: $$-2x + 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$$ Since $$\frac{3}{2} > 1$$, this x-intercept does not apply to the first piece. - For $$x \geq 1$$, $$2x - 1 = 0$$: $$2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}$$ However, $$\frac{1}{2} < 1$$, so this intercept only applies to the firstAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Intercepts
Graphing Functions
Domain and Range
Formulas
f(x) = -2x + 3 for x < 1
f(x) = 2x - 1 for x >= 1
Theorems
Piecewise Function Analysis
Intercept Calculation
Suitable Grade Level
Grades 9-11