Math Problem Statement

give me the range of 4x+5{x<2} 3x-8{x>2}

Solution

The given piecewise function is:

\begin{cases} 4x + 5 & \text{for } x < 2 \\ 3x - 8 & \text{for } x > 2 \end{cases}$$ To find the range of this function, we will analyze each piece. 1. **For $$x < 2$$:** The expression is $$4x + 5$$. As $$x$$ approaches negative infinity, $$4x + 5$$ will also approach negative infinity. At $$x = 2$$, the expression equals: $$4(2) + 5 = 8 + 5 = 13$$ However, since this expression is defined only for $$x < 2$$, it does not include $$x = 2$$. Thus, for this piece, the range is $$(-\infty, 13)$$. 2. **For $$x > 2$$:** The expression is $$3x - 8$$. As $$x$$ increases without bound, $$3x - 8$$ also increases without bound. At $$x = 2$$, the expression equals: $$3(2) - 8 = 6 - 8 = -2$$ However, since this expression is defined only for $$x > 2$$, it does not include $$x = 2$$. Thus, for this piece, the range is $$(-2, \infty)$$. ### Combined Range: - The first piece has the range $$(-\infty, 13)$$. - The second piece has the range $$(-2, \infty)$$. Since there is no overlap between these ranges, the combined range of the function is: $$(-\infty, 13) \cup (-2, \infty) = (-\infty, \infty)$$ Therefore, the **range** of the function is all real numbers $$\mathbb{R}$$. Would you like more details or clarification on any step? ### Related questions: 1. How do we find the domain of a piecewise function? 2. What are the continuity properties of this function at $$x = 2$$? 3. Can we graph the function and interpret the range visually? 4. How would the range change if one of the expressions were quadratic? 5. How do you solve piecewise functions with multiple conditions? ### Tip: Always check the endpoints and continuity of piecewise functions when determining the range and domain.

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Math Problem Analysis

Mathematical Concepts

Algebra
Piecewise Functions
Range of Functions

Formulas

f(x) = 4x + 5 for x < 2
f(x) = 3x - 8 for x > 2

Theorems

Piecewise function analysis

Suitable Grade Level

Grades 10-12

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