Chapter 8.1, Problem 58PPS

a.

To determine

Simplify the given expression.

Answer to Problem 58PPS

The required simplified expression is $x-1$ .

Explanation of Solution

Calculation:

Given expression $\frac{x^2-5x+4}{x-4}$

  $\begin{array}{l}=\frac{x^2-4x-x+4}{x-4}\text{ Factor}\\ =\frac{\left(x-4\right)\left(x-1\right)}{x-4}\text{ Common like terms}\\ =x-1\text{ Cancel like terms}\end{array}$

Thus, the required simplified expression is $x-1$

b.

To determine

Use a graphing calculator to make a table for both functions.

Answer to Problem 58PPS

The table of values is

  

Explanation of Solution

Calculation:

From part (a), $f\left(x\right)=\frac{x^2-5x+4}{x-4},g\left(x\right)=x-1$

Make a table of values as shown below:

  

Thus, the a table of values is

  

c.

To determine

Use a graphing calculator to make a table for both functions.

Answer to Problem 58PPS

Hence the $f\left(4\right)$ results in an error because the function is undefined at $x=4$

Explanation of Solution

Calculation:

From part (b),

  $f\left(4\right)=\text{Undefined},g\left(4\right)=3$

Thus, the $f\left(4\right)$ results in an error because the function is undefined at $x=4$

d.

To determine

Compare and contrast the graphs.

Answer to Problem 58PPS

Hence the graphs are almost identical, the only difference is that the graph of $f\left(x\right)$ has a hole at $x=4$ while the graph of $g\left(x\right)$ has none.

Explanation of Solution

Calculation:

From part (a), $f\left(x\right)=\frac{x^2-5x+4}{x-4},g\left(x\right)=x-1$

  

Thus the graphs are almost identical, the only difference is that the graph of $f\left(x\right)$ has a hole at $x=4$ while the graph of $g\left(x\right)$ has none.

e.

To determine

What conclusions can you draw about the expressions and the functions.

Answer to Problem 58PPS

Thus the expressions and functions are equivalent except for when $x=4$

Explanation of Solution

Calculation:

Clearly From the above parts,

Thus the expressions and functions are equivalent except for when $x=4$