Question

Answer the following questions regarding the function

$f\left(x\right)=\frac{x+5}{x-1}$

(a) What is the domain of f ?

(b) Is the point $(2,6)$on the graph of f ?

(c) If x = 3, what is $f\left(x\right)$? What point is on the graph of f ?

(d) If $f\left(x\right)=9$, what is x? What point is on the graph of f ?

(e) Is f a polynomial or rational function?

Short answer

a) Domain$=(-\infty ,1)\cup (1,\infty )$

b)No. The point does not lie on the graph of the function

c) The value is $f\left(x\right)=7$And the coordinate is $(3,7)$

d) The value is$x=\frac{7}{4}$. The coordinate is$(\frac{7}{4},9)$

e)The given function is a rational funtion.

Step-by-step solution

Step 1:Given information

We are given a function$f\left(x\right)=\frac{x+5}{x-1}$

Part a) Step 1: Find the domain

Domain is the set of all values for which a function exists.

Our given function does not exist when $$\begin{gathered} x-1=0 \\ x=1 \end{gathered}$$

Hence the domain of the function is$(-\infty ,1)\cup (1,\infty )$

Part b) Step 1: Check whether the point (2,6) lie of the graph of given function

The given point lies on the graph if it satisfies the given equation.

For that we substitute $x=2$in the function we get

$$\begin{gathered} f\left(x\right)=\frac{x+5}{x-1} \\ f\left(2\right)=\frac{2+5}{2-1} \\ f\left(2\right)=\frac{7}{1} \\ f\left(2\right)=7 \end{gathered}$$

Hence the coordinate is $(2,7)$

Hence this point does not lie on the graph of the function.

Part c) Step 1: Find f(3)

We get,

$$\begin{gathered} f\left(x\right)=\frac{x+5}{x-1} \\ f\left(3\right)=\frac{3+5}{3-1} \\ f\left(3\right)=\frac{8}{2} \\ f\left(3\right)=4 \end{gathered}$$

The coordinate can be given as$(3,4)$

Part d) Step 1: Find the value of x by substituting f(x)=9

We have,

$$\begin{gathered} f\left(x\right)=\frac{x+5}{x-1} \\ 9=\frac{x+5}{x-1} \\ 9(x-1)=x+5 \\ 9x-9=x+5 \\ 8x=14 \\ x=\frac{14}{8} \\ x=\frac{7}{4} \end{gathered}$$

The value of x for which$f\left(x\right)=9$is$x=\frac{7}{4}$

Part e) Step 1: Explanation

Given function is of type $\frac{A\left(x\right)}{B\left(x\right)}$where $A\left(x\right),B\left(x\right)$is a polynomial function and also $B\left(x\right)\ne 0$. Hence the given function is a rational function

Conclusion

a) Domain$=(-\infty ,1)\cup (1,\infty )$

b) NO. the point does not lie on the graph

c) The value is$f\left(x\right)=7$. The coordinate is$(3,7)$

d) The value is $x=\frac{7}{4}$. Also the coordinate is $(\frac{7}{4},9)$

e) given function is a rational function