Piecewise Function f(x) = 5x

Math Problem Statement

given f(x)= 5x-3 if x<2 and x^(2)+1 if x>=2 answer the following questions a. f(-1)= b. f(4) c. f(2)

Let's break down the function and solve the questions:

We have a piecewise function:

\begin{cases} 5x - 3 & \text{if } x < 2 \\ x^2 + 1 & \text{if } x \geq 2 \end{cases}$$ ### a. $$f(-1)$$ Since $$-1 < 2$$, we use the first case of the piecewise function: $$f(x) = 5x - 3$$ Substitute $$x = -1$$: $$f(-1) = 5(-1) - 3 = -5 - 3 = -8$$ So, $$f(-1) = -8$$. ### b. $$f(4)$$ Since $$4 \geq 2$$, we use the second case of the piecewise function: $$f(x) = x^2 + 1$$ Substitute $$x = 4$$: $$f(4) = 4^2 + 1 = 16 + 1 = 17$$ So, $$f(4) = 17$$. ### c. $$f(2)$$ Since $$2 \geq 2$$, we use the second case of the piecewise function: $$f(x) = x^2 + 1$$ Substitute $$x = 2$$: $$f(2) = 2^2 + 1 = 4 + 1 = 5$$ So, $$f(2) = 5$$. ### Final answers: - a. $$f(-1) = -8$$ - b. $$f(4) = 17$$ - c. $$f(2) = 5$$ Would you like a detailed explanation for any part of this? Here are some related questions to deepen understanding: 1. How would you handle a piecewise function where the boundaries are at different points? 2. What happens if the function is continuous at the boundary, such as at $$x = 2$$? 3. How do you determine which case of a piecewise function to use when calculating a value for $$f(x)$$? 4. What if the function was defined as $$5x + 3$$ for $$x \geq 2$$? How would that change the result for $$f(2)$$? 5. Can you graph a piecewise function, and how do you mark the points where the cases change? **Tip**: When working with piecewise functions, always check the inequalities carefully to select the right function piece based on the value of $$x$$.

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Piecewise Functions
Algebra
Substitution

f(x) = 5x - 3 for x < 2
f(x) = x^2 + 1 for x >= 2

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