Math Problem Statement
Two piecewise functions are shown below. What is the value of 3h(2) + 4g(1)?
Solution
To solve for , let’s evaluate each function individually based on the given piecewise definitions.
Step 1: Evaluate
The function is defined as:
-3x, & x < 2 \\ 4x + 1, & x \geq 2 \end{cases}$$ Since $$x = 2$$ falls into the $$x \geq 2$$ case, we use $$h(x) = 4x + 1$$. $$h(2) = 4(2) + 1 = 8 + 1 = 9$$ ### Step 2: Evaluate $$g(1)$$ The function $$g(x)$$ is defined as: $$g(x) = \begin{cases} x^2 + 2, & x < 3 \\ x^3, & x \geq 3 \end{cases}$$ Since $$x = 1$$ falls into the $$x < 3$$ case, we use $$g(x) = x^2 + 2$$. $$g(1) = (1)^2 + 2 = 1 + 2 = 3$$ ### Step 3: Substitute into $$3h(2) + 4g(1)$$ Now that we have $$h(2) = 9$$ and $$g(1) = 3$$, substitute these values: $$3h(2) + 4g(1) = 3(9) + 4(3)$$ $$= 27 + 12 = 39$$ ### Final Answer The value of $$3h(2) + 4g(1)$$ is **39**. Would you like further clarification on this solution? --- Here are five related questions to extend your understanding: 1. How would the answer change if $$x = 3$$ was used in $$g(x)$$? 2. What is the effect of using a different coefficient in front of $$h(2)$$ in this problem? 3. How do piecewise functions affect the continuity of a function? 4. Can you graph $$h(x)$$ and $$g(x)$$ to see their behavior visually? 5. What happens to the solution if we change $$h(x)$$ for $$x \geq 2$$ to a different linear function? **Tip:** When working with piecewise functions, always check which condition applies to the value of $$x$$ you're evaluating to avoid errors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise Functions
Function Evaluation
Basic Algebra
Formulas
For h(x): h(x) = -3x if x < 2, h(x) = 4x + 1 if x ≥ 2
For g(x): g(x) = x^2 + 2 if x < 3, g(x) = x^3 if x ≥ 3
3h(2) + 4g(1)
Theorems
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Suitable Grade Level
Grades 9-11