Advanced Functions 12
Advanced Functions 12
1st Edition
Chris Kirkpatrick, Kristina Farentino, Susanne Trew
ISBN: 9780176678326
Textbook solutions

All Solutions

Section 1-6: Piecewise Functions

Exercise 1
Step 1
1 of 3
#### (a)Exercise scan
#### (b)Exercise scan
Step 2
2 of 3
#### (c)Exercise scan
#### (d)Exercise scan
Step 3
3 of 3
#### (e)Exercise scan
#### (f)Exercise scan
Exercise 2
Step 1
1 of 1
Solution of this task is based on graphs of functions in previous task.

#### (a)

$f(x)$ $textbf{is discontinuous at}$ $x=1$

#### (b)

$f(x)$ $textbf{is discontinuous at}$ $x=0$

#### (c)

$f(x)$ $textbf{is discontinuous at}$ $x=-2$

#### (d)

$f(x)$ $textbf{is discontinuous at}$ $x=1$

#### (e)

$f(x)$ $textbf{is discontinuous at}$ $x=0$

#### (f)

$f(x)$ $textbf{is discontinuous at}$ $x=1$

Exercise 3
Step 1
1 of 2
The solution of this task are following functions:

#### (a)

$$
f(x)=begin{cases}
x^2-2 &, xleq1\
x+1 &, x>1\
end{cases}
$$

#### (b)

$$
f(x)=begin{cases}
left|x+1 right| &, x<1\
sqrt{x}+1 &, xgeq1\
end{cases}
$$

Result
2 of 2
#### (a)

$$
f(x)=begin{cases}
x^2-2 &, xleq1\
x+1 &, x>1\
end{cases}
$$

#### (b)

$$
f(x)=begin{cases}
left|x+1 right| &, x<1\
sqrt{x}+1 &, xgeq1\
end{cases}
$$

Exercise 4
Step 1
1 of 1
#### (a)

$textbf{The domain}$ of this function is $D=Bbb{R}$.

This function is $textbf{discontinuous}$ because all the pieces of
the function not join together at the endpoints of the given intervals.

#### (b)

$textbf{The domain}$ of this function is $D=Bbb{R}$.

This function is $textbf{continuous}$ because all the pieces of
the function join together at the endpoints of the given intervals.

Exercise 5
Step 1
1 of 4
#### (a)

This function is $textbf{discontionuous}$ because all the pieces of
the function not join together at the endpoints of the given intervals.

$textbf{The domain}$ of this function is $D=Bbb{R}$ and $textbf{the range}$ is $R=left{2,3 right}$.

$textbf{The graph}$ of this function is on the following picture:

Exercise scan

Step 2
2 of 4
#### (b)

This function is $textbf{contionuous}$ because all the pieces of
the function join together at the endpoints of the given intervals.

$textbf{The domain}$ of this function is $D=Bbb{R}$ and $textbf{the range}$ is $R=left[0,infty right)$.

$textbf{The graph}$ of this function is on the following picture:

Exercise scan

Step 3
3 of 4
#### (c)

This function is $textbf{contionuous}$ because all the pieces of
the function join together at the endpoints of the given intervals.

$textbf{The domain}$ of this function is $D=Bbb{R}$ and $textbf{the range}$ is $R=left[1,infty right)$.

$textbf{The graph}$ of this function is on the following picture:

Exercise scan

Step 4
4 of 4
#### (b)

This function is $textbf{contionuous}$ because all the pieces of
the function join together at the endpoints of the given intervals.

$textbf{The domain}$ of this function is $D=Bbb{R}$ and $textbf{the range}$ is $R=left[1,5 right]$.

$textbf{The graph}$ of this function is on the following picture:

Exercise scan

Exercise 6
Step 1
1 of 1
A function that describes
Graham’s total long-distance charge in terms of the number of long
distance minutes he uses in a month is:

$$
f(x)=begin{cases}
15 &, xleq500\
0.02&, x>500\
end{cases}
$$

Exercise 7
Step 1
1 of 2
A piecewise function that models this situation is:

$$
f(x)=begin{cases}
0.35x &, 0leq{x}leq100000\
0.45x-10000 &, 100000500000\
end{cases}
$$

Result
2 of 2
$$
f(x)=begin{cases}
0.35x &, 0leq{x}leq100000\
0.45x-10000 &, 100000500000\
end{cases}
$$
Exercise 8
Step 1
1 of 1
$textbf{The value of $k$}$ for which is the following function continuous is for $k=4$.

Here we have $textbf{the graph}$ of this function, for $k=4$:

Exercise scan

Exercise 9
Step 1
1 of 3
#### (a)

Here we have $textbf{the graph}$ of this function:

Exercise scan

Step 2
2 of 3
#### (b)

This function is $textbf{discontinuous}$ at $x=6$.

#### (c)

We will calculate $2^x-(4x+8)$ at $x=6$.

$2^6-(4cdot6+8)=64-32=32$ fishes.
#### (d)

Using the function that represents the time after the spill

$4x+8=64$

$4x=56$

$x=14$

#### (e)

For example: $3$ possible events are environmental chages, introduction of a new predator and increased fishing.

Result
3 of 3
see solution
Exercise 10
Step 1
1 of 2
Answers may vary.For example:Exercise scan
Result
2 of 2
see solution
Exercise 11
Step 1
1 of 1
$$
f(x)=begin{cases}
x+3 &, xgeq-3\
-(x+3) &, x<-3\
end{cases}
$$

And,here is the graph which follows this function:

Exercise scan

Exercise 12
Step 1
1 of 1
This function is $textbf{continuous}$,because all the pieces of
the function join together at the endpoints of the given intervals.We can see that from the following graph of this function:

Exercise scan

Exercise 13
Step 1
1 of 3
$textbf{This piecewise function}$ is actually the following function:

$$
f(x)=begin{cases}
0 &, 0leq{x}<10\
10 &, 10leq{x}<20\
20 &, 20leq{x}<30\
30 &, 30leq{x}<40\
40 &, 40leq{x}<50\
end{cases}
$$

And, $textbf{the graph}$ of this function is on the following picture:

Exercise scan

Step 2
2 of 3
This function has got this name precisely because of the layout of its graphics. In fact, this graph has jumps of the same size, so it reminds us of the stairs, becauseof that, the graphics that look like this have got such a name.
Result
3 of 3
$$
f(x)=begin{cases}
0 &, 0leq{x}<10\
10 &, 10leq{x}<20\
20 &, 20leq{x}<30\
30 &, 30leq{x}<40\
40 &, 40leq{x}<50\
end{cases}
$$
Exercise 14
Step 1
1 of 1
Adding or subtracting factors to the function $y = x$ makes this function for the value of $k$ move up or down by $y$-axis. The graph shows that even for one function $y = x$ can not be simultaneously connected to the functions $y=5x$ and $y=2x^2$ , so it can not be continuous for one $k$.

Exercise scan

Exercise 15
Step 1
1 of 1
$textbf{The graph}$ of this function is on the following picture:

Exercise scan

Exercise 16
Step 1
1 of 2
#### (a)

For example, for this task we can use $textbf{following function}$ which is made of three parent functions, $y=e^x$, $y=sqrt{x}$ and $y=x$.

$$
f(x)=begin{cases}
e^x &, -4leq{x}<0\
sqrt{x} &, 0leq{x}<4\
x &, {x}geq4\
end{cases}
$$

#### (b)

Here we have $textbf{the graph}$ of the function we created at $(a)$:

Exercise scan

Step 2
2 of 2
#### (c)

This function is $textbf{discontinuous}$ because all the pieces of
the function not join together at the endpoints of the given intervals.We can see that from previous graph of this function.

#### (d)

We can transform function we made at $(a)$ in the following function and by this transformation we will make continious function:

$$
f(x)=begin{cases}
e^x-1 &, -4leq{x}<0\
sqrt{x} &, 0leq{x}<4\
x-2 &, {x}geq4\
end{cases}
$$

We can see and from the following graph of this transformed function that it is $textbf{continuous}$:

Exercise scan

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