About Piecewise-Defined Functions:
Piecewise-Defined functions have different rules or definitions over different intervals of the domain. Common examples include the absolute value function along with the greatest integer function or floor function. We will also need to graph piecewise-defined functions along with evaluating piecewise-defined functions for given values of x.
Test Objectives
- Demonstrate the ability to evaluate a piecewise-defined function for given values of x
- Demonstrate the ability to sketch the graph of a piecewise-defined function
#1:
Instructions: for the given piecewise-defined function, find f(-3) and f(3).
$$a)\hspace{.2em}$$ $$\ f(x)=\begin{cases}-5x-3 & \text{if}\hspace{.2em}x ≥ 0 \\ x^2 - 3 & \text{if}\hspace{.2em}x < 0 \end{cases}$$
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#2:
Instructions: for the given piecewise-defined function, find f(-5), f(1), and f(7).
$$a)\hspace{.2em}$$ $$\ f(x)=\begin{cases}(x - 2)^3 + 1 & \text{if}\hspace{.2em}x ≥ 2 \\ x^4 - x - 5 & \text{if}\hspace{.2em}0 ≤ x < 2 \\ -2x^2 - 7 & \text{if}\hspace{.2em}x < 0 \end{cases}$$
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#3:
Instructions: find f(-2.3) and f(7.6).
$$a)\hspace{.2em}$$ $$f(x)=[x]$$
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#4:
Instructions: graph the given piecewise-defined function.
$$a)\hspace{.2em}$$ $$\ f(x)=\begin{cases}\sqrt{x}& \text{if}\hspace{.2em}x ≥ 0 \\ |x + 2| & \text{if}\hspace{.2em}x < 0 \end{cases}$$
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#5:
Instructions: graph the given piecewise-defined function.
$$a)\hspace{.2em}$$ $$\ f(x)=\begin{cases}x - 4 & \text{if}\hspace{.2em}x ≥ 2 \\ 3 & \text{if}\hspace{.2em}{-}2 < x < 2 \\ 2x - 1 & \text{if}\hspace{.2em}x ≤ {-}2 \end{cases}$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}$$ $$ f(3)=-18 $$ $$f(-3)=6$$
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#2:
Solutions:
$$a)\hspace{.2em}$$ $$ f(7)=126$$ $$ f(1)=-5 $$ $$ f(-5)=-57$$
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#3:
Solutions:
$$a)\hspace{.2em}$$ $$f(-2.3)=-3$$ $$f(7.6)=7$$
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#4:
Solutions:
$$a)\hspace{.2em}$$
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#5:
Solutions:
$$a)\hspace{.2em}$$
