Math Problem Statement
Solve
The function f is defined as f(x) = { -2x + 3 if x < 1, 4x - 3 if x ≥ 1. Solve for the domain, intercepts, graph the function, and find the range.
Solution
Let's analyze the piecewise-defined function given in the image and solve the questions step by step:
The function is defined as:
\begin{cases} -2x + 3 & \text{if } x < 1 \\ 4x - 3 & \text{if } x \geq 1 \end{cases}$$ ### (a) Find the domain of the function. The domain of the function includes all values of $$x$$ for which the function is defined. In this case, the function is defined for all real numbers: - The first piece, $$-2x + 3$$, is valid for $$x < 1$$. - The second piece, $$4x - 3$$, is valid for $$x \geq 1$$. Thus, the domain of $$f(x)$$ is all real numbers: $$\boxed{(-\infty, \infty)}$$ ### (b) Locate any intercepts. #### x-intercepts: To find the x-intercepts, set $$f(x) = 0$$ and solve for $$x$$. 1. For $$x < 1$$: $$-2x + 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2} = 1.5$$ Since $$x = 1.5$$ does not satisfy $$x < 1$$, there is no x-intercept for this piece of the function. 2. For $$x \geq 1$$: $$4x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{4}$$ Since $$x = 0.75$$, this gives the x-intercept. Thus, the x-intercept is: \[{boxAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Linear Equations
Domain and Range
Intercepts
Graphing
Formulas
y = mx + b (Equation of a line)
Theorems
Linear Function Properties
Domain and Range Analysis
Suitable Grade Level
Grades 9-12