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Lectures
3 Videos

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2.03mins

Introduction to Piece Wise Function

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2.04mins

Piece Wise Function ex2

Different Types of Discontinuity

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Different Types of Discontinuity

Solutions
23 Videos

What does the following piecewise function look like? Show your work.

```
f(x) =\begin{cases}2, &\text{if } x < 1\\ 3x, &\text{if } x \geq 1 \end{cases}
```

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0.39mins

Q1a

What does the following piecewise function look like? Show your work.

```
f(x) =\begin{cases}-2x, &\text{if } x < 0\\ x + 4, &\text{if } x \geq 0 \end{cases}
```

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0.51mins

Q1b

What does the following piecewise function look like? Show your work.

```
f(x) =\begin{cases}|x|, &\text{if } x \leq -2\\ -x^2, &\text{if } x > -2 \end{cases}
```

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Q1c

What does the following piecewise function look like? Show your work.

```
f(x) =\begin{cases} \phantom{+} |x + 2|, &\text{if } x \leq -1\\ -x^2 + 2, &\text{if } x >-1 \end{cases}
```

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Q1d

What does the following piecewise function look like? Show your work.

```
f(x) =\begin{cases} \sqrt{x}, &\text{if } x < 4\\2^x, &\text{if } x \geq 4 \end{cases}
```

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1.13mins

Q1e

What does the following piecewise function look like? Show your work.

```
f(x) =\begin{cases} \phantom{+}\frac{1}{x}, &\text{if } x < 1\\-x, &\text{if } x \geq 1 \end{cases}
```

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Q1f

Are the functions continuous? If not, where is it discontinuous? Show your work.

a) ```
f(x) =\begin{cases}2, &\text{if } x < 1\\ 3x, &\text{if } x \geq 1 \end{cases}
```

b) ```
f(x) =\begin{cases}-2x, &\text{if } x < 0\\ x + 4, &\text{if } x \geq 0 \end{cases}
```

c) ```
f(x) =\begin{cases}|x|, &\text{if } x \leq -2\\ -x^2, &\text{if } x > -2 \end{cases}
```

d) ```
f(x) =\begin{cases} \phantom{+} |x + 2|, &\text{if } x \leq -1\\ -x^2 + 2, &\text{if } x >-1 \end{cases}
```

e) ```
f(x) =\begin{cases} \sqrt{x}, &\text{if } x < 4\\2^x, &\text{if } x \geq 4 \end{cases}
```

f) ```
f(x) =\begin{cases} \phantom{+}\frac{1}{x}, &\text{if } x < 1\\-x, &\text{if } x \geq 1 \end{cases}
```

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Q2

What is the algebraic representation of the piecewise function? Use function notation. Show your work.

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Q3a

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Q3b

What is the domain of each piecewise function below? Is the function continuous? If not, where is it discontinuous?

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1.09mins

Q4

What does the following piecewise function look like?

```
f(x) =
\begin{cases}2, &\text{if } x < -1 \\ 3, &\text{if } x \geq -1\end{cases}
```

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Q5a

Sketch `f(x)`

.

```
f(x) =
\begin{cases}-x, &\text{if } x \leq 0 \\ x, &\text{if } x > 0\end{cases}
```

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Q5b

What does the following piecewise function look like?

```
f(x) =
\begin{cases}x^2 + 1, \text{if } x < 2 \\ 2x+1, \text{if } x \geq 2\end{cases}
```

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Q5c

What does the following piecewise function look like?

```
f(x) =
\begin{cases} 1, &\text{if } x < -1 \\ x+2, &\text{if } -1\leq x \leq 3 \\ 5, &\text{if } x> 3\end{cases}
```

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Q5d

Graham's long-distance telephone plan includes the first `500`

min per A month in the `\$15.00`

monthly charge. For each minute after `500`

min, Graham is charged `\$0.02`

. What is a function that describes Graham's total long-distance charge in terms of the number of long distance minutes he uses in a month? Show your work.

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Q6

Many income tax systems are calculated using a tiered method. Under a certain tax law, the first `\$100 000`

of earnings are subject to a `35\%`

tax; earnings greater than `\$100 000`

and up to `\$500 000`

are subject to a `45\%`

tax. Any earnings greater than `\$500 000`

are taxed at `55\%`

. What is a piecewise function models this situation? Show your work clearly.

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Q7

What is the value of `k`

so that the following function is continuous? Show your work.

```
f(x) = \begin{cases} x^2-k, \text{if }x < -1 \\ 2x - 1, \text{if }x \geq -1 \end{cases}
```

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Q8

The fish population, in thousands, in a lake at any time, `x`

, in years is modelled by the following function:

```
f(x) = \begin{cases} 2^x, &\text{if } 0\leq x \leq 6 \\ 4x + 8, &\text{if } x > 6\end{cases}
```

**(a)** How many fish were killed by the chemical spill? Notice the drop in the value of `f(x)`

is where chemical spill happens.

**(b)** At what time did the population recover to the level it was before
the chemical spill?

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Q9

An absolute value function can be written as a piecewise function that involves two linear functions. What is the equation of the function `f(x) = |x + 3|`

as a piecewise function? Show your work.

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Q11

The demand for a new CD is described by

```
\displaystyle
D(p) = \begin{cases} \frac{1}{p^2}, &\text{if } 0 < p \leq 15 \\ 0, &\text{if }x > 15 \end{cases}
```

where `D`

is the demand for the CD at price `p`

, in dollars.
Find where the demand function is discontinuous. Show your work.

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Q12

Consider a function, `f(x)`

, that takes an element of its domain and rounds it down to the nearest 10. Thus, `f(15.6) =10`

, while `f(21.7) = 20`

and `f(30) = 30`

. Write the piecewise function. You may limit the domain to `x\in[0, 50)`

.

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Q13

What value of `k`

will make the following function continuous? Explain.

```
f(x) = \begin{cases} 5x, &\text{if }x < -1 \\x + k, &\text{if } -1\leq x \leq 3 \\ 2x^2, &\text{if }x > 3 \end{cases}
```

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Q14

The greatest integer function is a step function that is written as `f(x) = [x]`

where `f(x)`

is the greatest integer less than or equal to x. In other words, the greatest integer function rounds any number down to the nearest integer. For example, the greatest integer less than or equal to the number `[5.3] = 5`

, while the greatest integer less than or equal to the number `[-5,3] = -6`

. What does the graph of `f(x) = [x]`

look like? Show your work.

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Q15