#### To determine

**To find:** To find the domain and range of the function f(x)=lnx-1-1.

#### Answer

The domain of the function is (1 ,∞) and the range of the function is (-∞,∞).

#### Explanation

**Calculation:**

**Step 1:** First we will find the domain of y=lnx-1-1. We know that domain

includes all values of x for which the function is defined.

**Step 2:** We know that the function f(x)=lnx-1-1 is defined if and only if (x-1)>0

this implies x>1 and that gives the domain (1 ,∞).

**Step 3:** Now, we will find the range of the function fx=lnx-1-1.

We know that the range include all values of *f(x)* for which there is some *x* such that fx=

lnx+2. The range of the f(x)=lnx-1-1 is all the real number (-∞,∞).

**Conclusion:**

The domain of the function is (1 ,∞) and the range of the function is (-∞,∞).

#### To determine

**To find:** x*-* intercept of the graph of f.

#### Answer

x*-* intercept of the graph of f is ≈3.718.

#### Explanation

**Calculation:** To find the *x-*intercept of the graph of *f*, we have to find the value of x for which

fx=0 that is

lnx-1-1=0

or ln(x-1)=1.

Now, we take the exponential function of both sides which is the inverse function of the

logarithmic function. Thus

eln(x-1)=e1

Since we know that

elnx=x (1)

From equation (1) we have

x-1=e1 or x=e+1 Now using scientific calculator, we can calculate the value of x up to 3

decimal places. So, the approximate value of x-intercept up to 3 decimal places is x≈3.718.

**Conclusion:**

x*-* intercept of the graph of f is ≈3.718.

#### To determine

**To find:** Sketch the graph of f.

#### Answer

See the graph

#### Explanation

To sketch the graph of f(x)=lnx-1-1, first we draw the graph of f(x)=lnx( as given in

Figure 3)

and then using the transformation gx=fx-1. Thus, we will get the graph of fx=

ln(x-1), by shifting the graph of f(x)=lnx, 1 unit right. So, the graph of fx=ln(x-1) is

Now by using the transformation g(x)=f(x)-1 and by shifting the graph of fx=ln(x-1)

1 unit downward, we get the graph of fx=lnx-1-1. Therefore, the graph of fx=

ln(x-1)-1 is

**Conclusion:**

See the graph