Math Problem Statement

Let g be the function defined by: g(x) = - (1/2)x + 5 if x < 6 sqrt(x - 6) if x >= 6 Find g(-6), g(0), g(6), and g(12).

Solution

We are given a piecewise function g(x)g(x) defined as follows:

\begin{cases} -\frac{1}{2}x + 5 & \text{if } x < 6 \\ \sqrt{x - 6} & \text{if } x \geq 6 \end{cases}$$ We need to evaluate this function at $$g(-6), g(0), g(6),$$ and $$g(12)$$. ### Step-by-step Calculation: 1. **For $$g(-6)$$:** Since $$-6 < 6$$, we use the first part of the piecewise function: $$g(-6) = -\frac{1}{2}(-6) + 5 = 3 + 5 = 8$$ 2. **For $$g(0)$$:** Since $$0 < 6$$, we again use the first part of the piecewise function: $$g(0) = -\frac{1}{2}(0) + 5 = 0 + 5 = 5$$ 3. **For $$g(6)$$:** Since $$6 \geq 6$$, we use the second part of the piecewise function: $$g(6) = \sqrt{6 - 6} = \sqrt{0} = 0$$ 4. **For $$g(12)$$:** Since $$12 \geq 6$$, we use the second part of the piecewise function: $$g(12) = \sqrt{12 - 6} = \sqrt{6}$$ Approximating $$\sqrt{6}$$, we get: $$g(12) \approx 2.449$$ ### Final Answers: - $$g(-6) = 8$$ - $$g(0) = 5$$ - $$g(6) = 0$$ - $$g(12) \approx 2.449$$ Would you like more details or clarifications on any of the steps? --- Here are 5 related questions for further practice: 1. What is the domain of the function $$g(x)$$? 2. Evaluate $$g(10)$$ using the same function. 3. How does the function $$g(x)$$ behave as $$x \to -\infty$$? 4. Find the inverse of $$g(x)$$ for $$x \geq 6$$. 5. What is the range of the function $$g(x)$$? **Tip:** When dealing with piecewise functions, always check which part of the function applies based on the value of $$x$$.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Linear Functions
Square Root Functions

Formulas

Linear equation: g(x) = -1/2 * x + 5
Square root function: g(x) = sqrt(x - 6)

Theorems

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Suitable Grade Level

Grades 9-12