Chapter 2.1, Problem 34PPS

a.

To determine

Graph each function on separate screen.

Answer to Problem 34PPS

  $f\left(x\right)=x^2$

  

  $g\left(x\right)=2^x$

  

  $h\left(x\right)=x^3-3x^2-5x+6$

  

  $j\left(x\right)=x^3$

  

Explanation of Solution

Given information:

  $\begin{array}{l}f\left(x\right)=x^{2}g\left(x\right)=2^x\\ h\left(x\right)=x^3-3x^2-5x+6j\left(x\right)=x^3\end{array}$

Calculation:

Consider the given one by one,

Use $TI-83$ calculator to graph the function, $f\left(x\right)=x^2$

Step 1. Press $Y=$ and put the function on $Y1$

  

Step2.Press GRAPH and TRACE button,

  

Use $TI-83$ calculator to graph the function, $g\left(x\right)=2^x$

Step 1. Press $Y=$ and put the function on $Y1$

  

Step 2. Press GRAPH and TRACE button,

  

Use $TI-83$ calculator to graph the function, $h\left(x\right)=x^3-3x^2-5x+6$

Step 1. Press $Y=$ and put the function on $Y1$

  

Step 2. Press GRAPH and TRACE button,

  

Use $TI-83$ calculator to graph the function, $j\left(x\right)=x^3$

Step 1. Press $Y=$ and put the function on $Y1$

  

Step 2. Press GRAPH and TRACE button,

  

.

b.

To determine

Tabulate the graph of each function to show intersection point on horizontal line.

Answer to Problem 34PPS

  $\begin{array}{cc}Function& Possible\mathrm{int}er\sec tionpo\mathrm{int}s\\ f\left(x\right)=x^2& 0,1,2\\ g\left(x\right)=2^x& 0,1\\ h\left(x\right)=x^3-3x^2-5x+6& 1,2,3\\ j\left(x\right)=x^3& 1\end{array}$

Explanation of Solution

Given information:

Graphs shown in part $\left(a\right)$

Calculation:

Consider the graphs of part $\left(a\right)$ ,

  $\begin{array}{cc}Function& Possible\mathrm{int}er\sec tionpo\mathrm{int}s\\ f\left(x\right)=x^2& 0,1,2\\ g\left(x\right)=2^x& 0,1\\ h\left(x\right)=x^3-3x^2-5x+6& 1,2,3\\ j\left(x\right)=x^3& 1\end{array}$

c.

To determine

Identify the function for given condition.

Answer to Problem 34PPS

  $g\left(x\right)$ and $j\left(x\right)$ are one-to −one function.

  $f\left(x\right)$ and $h\left(x\right)$ are not one-to −one function.

Explanation of Solution

Given information:

For a function to be one-to-one, a horizontal line can intersect the function at most once.

Calculation:

It is not easy to check the function of all graphs, introduce a pencil on all graph if it is parallel to the $x-axis$ and then move the pencil up and down then check for the graph lines intersect with the pencil , if they intersect then graph is one-to-one.

  $g\left(x\right)$ and $j\left(x\right)$ meet the pencil line one time, therefore they are one-to −one,

  $f\left(x\right)$ and $h\left(x\right)$ functions are one-to −one.

Hence, $g\left(x\right)$ and $j\left(x\right)$ are one-to −one, $f\left(x\right)$ and $h\left(x\right)$ functions are one-to −one.

d.

To determine

Identify the function for given condition.

Answer to Problem 34PPS

  $h\left(x\right)$ and $j\left(x\right)$ are onto functions,

  $f\left(x\right)$ and $g\left(x\right)$ are not onto functions.

Explanation of Solution

Given information:

For a function to be onto, a horizontal line can intersect the function at least once.

Calculation:

A parallel line on the graph of $h\left(x\right)$ and $j\left(x\right)$ will cross over the graph lines more than one at a time therefore they are onto functions, $f\left(x\right)$ and $g\left(x\right)$ does not intersect the graph lines. They are not onto functions.

e.

To determine

Create the table of given functions.

Answer to Problem 34PPS

  $\begin{array}{ccc}Functions& one-to-one& Onto\\ f\left(x\right)=x^2& No& No\\ g\left(x\right)=2^x& Yes& No\\ h\left(x\right)=x^3-3x^2-5x+6& No& Yes\\ j\left(x\right)=x^3& Yes& Yes\end{array}$

Explanation of Solution

Given information:

One-to one function and onto function.

Calculation:

  $\begin{array}{ccc}Functions& one-to-one& Onto\\ f\left(x\right)=x^2& No& No\\ g\left(x\right)=2^x& Yes& No\\ h\left(x\right)=x^3-3x^2-5x+6& No& Yes\\ j\left(x\right)=x^3& Yes& Yes\end{array}$