1.5E: Piecewise-Defined Functions (Exercises)
Section 1.5 Exercises
1. Let \( f(x)=\left\{\begin{array}{ccc} {x+1} & {if} & {x<2} \\ {5} & {if} & {x\ge 2} \end{array}\right. \)
a) Evaluate \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\)
b) Sketch a graph of the piecewise defined function.
2. Let \( f(x=\left\{\begin{array}{ccc} {4} & {if} & {x<0} \\ {\sqrt{x} } & {if} & {x\ge 0} \end{array}\right.\)
a) Evaluate \(f(-1)\), \(f(0)\), \(f(1)\), and \(f(2)\)
b) Sketch a graph of the piecewise defined function.
3. Let \( f(x)=\left\{\begin{array}{ccc} {3x-1 } & {if} & {x<1} \\ {x+2} & {if} & {x\ge 1} \end{array}\right. \)
a) Evaluate \(f(0)\), \(f(1)\), \(f(2)\), and \(f(3)\)
b) Sketch a graph of the piecewise defined function.
4. Let \( f(x)=\left\{\begin{array}{ccc} {x^2} & {if} & {x<1} \\ {-2x+5 } & {if} & {x\ge 1} \end{array}\right.\)
a) Evaluate \(f(0)\), \(f(1)\), \(f(2)\), and \(f(3)\)
b) Sketch a graph of the piecewise defined function.
5. Sketch a graph of the piecewise defined function \( f(x)=\left\{\begin{array}{ccc} {3} & {if} & {x\le -2} \\ {-x+1} & {if} & {-2<x\le 1} \\ {x+2} & {if} & {x>1} \end{array}\right.\). Then identify the domain and range of \(f\).
6. Sketch a graph of the piecewise defined function \( f(x)=\left\{\begin{array}{ccc} {2x+1} & {if} & {x\le 0} \\ {x+1} & {if} & {0<x\le 2} \\ {3} & {if} & {x>2} \end{array}\right.\) Then identify the domain and range of \(f\).
7. Write a formula for the piecewise function graphed below.
a) b)
8. Write a formula for the piecewise function graphed below.
a) b)
9. Write a formula for the piecewise function graphed below.
a) b)
10. Suppose a commercial cloud data plan charges $5 per year plus $2.5 per GB for the first 10 GB and then $1.5 per GB for each additional GB of data. Find a piecewise defined formula for the yearly cost \(C\) in terms of the data load \(D\) in GB.
11. Suppose a painting company charges $2.40 per \(\text{ft}^2\) to paint a house of up to 2000 \(\text{ft}^2\). For houses with more than 2000 \(\text{ft}^2\) to paint, they charge $2.10 per \(\text{ft}^2\) for each additional square foot of area above 2000 \(\text{ft}^2\). Find a piecewise defined formula for the yearly cost \(C\) in terms of the square footage painted \(A\).
12. Suppose an author of a new book receives $30,000 to write it. For the first 50,000 copies sold, they will make $1.24 per copy sold. After that, they will make $1.02 per copy sold. Find a piecewise defined formula for the amount made \(A\) in terms of the number of book copies sold \(b\).
- Answer
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1. a) \(f(1)=2\), \(f(2)=5\), \(f(3)=5\), and \(f(4)=5\)
b)
3. a) \(f(0)=-1\) \(f(1)=3\) \(f(2)=4\) \(f(3)=4\)
b)
5.
7. a) \(f(x) = \begin{cases} 2 & if & -6 \le x \le -1 \\ -2 & if & -1 < x \le 2 \\ -4 & if & 2 < x \le 4 \end{cases}\)
b) \(f(x) = \begin{cases} -4 & if & -6 \le x \le -2 \\ 5 & if & -2 < x \le 1 \\ -3 & if & 1 < x \le 5 \end{cases}\)
9. a) \(f(x) = \begin{cases} 2x + 3 & if & -3 \le x < -1 \\ x - 1 & if & -1\le x \le 2 \\ -2 & if & 2 < x \le 5 \end{cases}\)
11. \(C =f(A) = \begin{cases} 2.40A & if & 0 \le A \le 2000 \\ 2.10A+600 & if & 2000 < A \end{cases}\)