Evaluate Piecewise Functions: Step-by-Step Solutions

Math Problem Statement

Direction: Answer the following in a 1 whole sheet of paper. Provide your complete solution.

I. Evaluate the following piecewise functions.

f(3) =

f(- 4) =

f(- 2) =

Let's evaluate the given piecewise function step by step. The function $f(x)$ is defined as:

x + 5 & \text{if } x < -2 \\ x^2 + 2x + 3 & \text{if } x \geq -2 \end{cases}$$ Now, let's find the values for $$f(3)$$, $$f(-4)$$, and $$f(-2)$$. ### 1. $$f(3)$$ Since $$3 \geq -2$$, we use the second piece of the function: $$f(x) = x^2 + 2x + 3$$ Substitute $$x = 3$$ into the formula: $$f(3) = 3^2 + 2(3) + 3 = 9 + 6 + 3 = 18$$ Thus, $$f(3) = 18$$. ### 2. $$f(-4)$$ Since $$-4 < -2$$, we use the first piece of the function: $$f(x) = x + 5$$ Substitute $$x = -4$$ into the formula: $$f(-4) = -4 + 5 = 1$$ Thus, $$f(-4) = 1$$. ### 3. $$f(-2)$$ Since $$-2 \geq -2$$, we use the second piece of the function: $$f(x) = x^2 + 2x + 3$$ Substitute $$x = -2$$ into the formula: $$f(-2) = (-2)^2 + 2(-2) + 3 = 4 - 4 + 3 = 3$$ Thus, $$f(-2) = 3$$. ### Final Answers: - $$f(3) = 18$$ - $$f(-4) = 1$$ - $$f(-2) = 3$$ --- Would you like more details or have any questions? Here are 5 related problems for practice: 1. Evaluate $$f(0)$$ using the same piecewise function. 2. Evaluate $$f(-3)$$ using the same piecewise function. 3. Find $$f(x)$$ for $$x = 1$$. 4. What is the value of $$f(x)$$ when $$x = -5$$? 5. For what values of $$x$$ does $$f(x) = 3$$? **Tip**: When working with piecewise functions, always check which part of the function applies based on the given value of $$x$$.

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