Piecewise Functions

1.  If   \( f(x)=\begin{cases} 2x-7,   \quad  x  \geq 0, \\  \sqrt{5-2x},  \quad  x  < 0,     \end{cases} \)  and  \(g(x)=\begin{cases}   x^2-x+1,  \quad x < 1, \\  |2x-3| , \quad   x \geq 1,  \end{cases} \) evaluate the following:  solution

(a) \(f(-2)\)     (b) \(g(-3)\)       (c) \(g(1)\)        (d) \(f(5)\)       (e)  \(g(0)\)        (f) \(g(3)\)

2. If  \(g(x)=\begin{cases} 2x+5,   \quad  x < 1\\ 4-2x, \quad  x \geq 1.    \end{cases} \) Find the following:  solution

 (a) \( g(0) \)           (b) \(g(-1)\)                 (c) \( g(5) \)

 (d)  sketch the graph of \(y=g(x)\) 

 (e)  the domain and range of \(g\) from the graph.

3. Suppose the tax liability \(T \) on \(x\) dollars of taxable income is given by

\(T(x)=\begin{cases}
0.06x \qquad \qquad \quad \text{if}  \, \, 0 < x < 30,000,\\
1,800+0.04x \qquad \text{if}  \, \, 30,000 \leq x < 80,000,\\
5,000+0.05x \qquad \text{if} \,  \, x \geq 80,000.
\end{cases}\)  

Find the tax liability on each taxable income.   solution

(a) \(\displaystyle{\$25,000}\)            (b) \(\displaystyle{\$125,000}\)            (c) \(\displaystyle{\$60,000}\)