Chapter 2.7, Problem 21E

To determine

To sketch the function $f(t)=\frac{1-2t}{t}$ , also find their intercepts, asymptotes and holes.

Answer to Problem 21E

  $t=0,y=-2$

Explanation of Solution

Given:

Function: $f(t)=\frac{1-2t}{t}$

Calculation for graph:

Consider $f(t)=\frac{1-2t}{t}$

Values of t	Values of f (t)
0	Infinity
1	-1
-1	-3
2	-1.5
-2	-2.5

By taking different values of t, the graph can be plotted.

Graph:

  

Calculation:

Intercepts:

Let $f(t)=y.$

  $\Rightarrow y=\frac{1-2t}{t}$

To find x intercepts, put y = 0,

  $\begin{array}{l}\Rightarrow y=0\\ \Rightarrow \frac{1-2t}{t}=0\\ \Rightarrow 1-2t=0\\ \Rightarrow 2t=1\\ \Rightarrow t=\frac{1}{2}\\ \Rightarrow t=0.5\end{array}$

So, x intercept is (0.5, 0)

To find y intercepts, put t = 0,

  $\begin{array}{l}\Rightarrow y=\frac{1-2t}{t}\\ \Rightarrow y=\frac{1-2(0)}{0}\\ \Rightarrow y=\frac{1-0}{0}\\ \Rightarrow y=\frac{1}{0}\to \text{undefined}\end{array}$

So, y intercept is not defined for given function.

Asymptotes:

Vertical asymptotes:

To find vertical asymptotes, put denominator of the given function equal to zero.

  $\Rightarrow t=0$

Horizontal asymptotes:

As the degree of numerator is equal to the degree of the denominator, the horizontal asymptote is

  $\begin{array}{l}\Rightarrow y=\frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}\\ \Rightarrow y=\frac{-2}{1}\\ \Rightarrow y=-\text{2}\end{array}$

Holes:

Here, the given function does not contain common factor in numerator and denominator. So, the function does not have any holes.