Math Problem Statement
Suppose that the function g is defined, for all real numbers, as follows: g(x) = { 1/4x + 2 if x ≤ -1 (x-1)^2 - 1 if -1 < x < 2 -1/4x + 1 if x ≥ 2 Find g(-1), g(1), and g(5).
Solution
We are given a piecewise function , and we need to evaluate it at three different points: , , and . The function is defined as:
\frac{1}{4}x + 2 & \text{if } x \leq -1 \\ (x - 1)^2 - 1 & \text{if } -1 < x < 2 \\ -\frac{1}{4}x + 1 & \text{if } x \geq 2 \end{cases}$$ Let's go step by step. ### 1. Evaluate $$g(-1)$$: Since $$-1 \leq -1$$, we use the first case: $$g(-1) = \frac{1}{4}(-1) + 2 = -\frac{1}{4} + 2 = \frac{7}{4}$$ ### 2. Evaluate $$g(1)$$: Since $$-1 < 1 < 2$$, we use the second case: $$g(1) = (1 - 1)^2 - 1 = 0^2 - 1 = -1$$ ### 3. Evaluate $$g(5)$$: Since $$5 \geq 2$$, we use the third case: $$g(5) = -\frac{1}{4}(5) + 1 = -\frac{5}{4} + 1 = \frac{-5}{4} + \frac{4}{4} = \frac{-1}{4}$$ ### Final Answers: $$g(-1) = \frac{7}{4}, \quad g(1) = -1, \quad g(5) = \frac{-1}{4}$$ Would you like more details or have any other questions? Here are 5 related questions to expand on the concept: 1. How would you describe the behavior of the function $$g(x)$$ at $$x = 2$$? 2. What is the domain and range of the function $$g(x)$$? 3. Can you graph the piecewise function and interpret the graph? 4. What happens to the values of $$g(x)$$ as $$x \to -\infty$$ and $$x \to \infty$$? 5. How would the function $$g(x)$$ change if the second interval were $$-1 \leq x < 2$$? **Tip**: For piecewise functions, always carefully check which interval your input falls into before evaluating.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Algebra
Formulas
g(x) = 1/4x + 2
g(x) = (x-1)^2 - 1
g(x) = -1/4x + 1
Theorems
Piecewise Function Evaluation
Suitable Grade Level
Grades 9-12