Question

(a) What are the domain and range of f?

(b) What is the x-intercept of the graph of f?

(c) Sketch the graph of f.

55. \(f\left( x \right) = {\bf{ln}}x + {\bf{2}}\)

Short answer

a) The domain is \(x > 0\) and the range is \(\mathbb{R}\).

(b) The x-intercept of the graph is \({e^{ - 2}}\).

(c) The graph is shown below:

Step-by-step solution

Find the domain and range of f

The function f is defined for \(x > 0\). Therefore the domain of the function is \(x > 0\) , and the range of the function is \(\mathbb{R}\).

Find the x-intercept of the graph

Substitute 0 for \(f\left( x \right)\) in the function \(f\left( x \right) = \ln x + 2\).

\(\begin{aligned}0 &= \ln x + 2\\\ln x &= - 2\\x &= {e^{ - 2}}\end{aligned}\)

Thus, the x-intercept is \({e^{ - 2}}\).

Sketch the graph of f

Shift the graph of \(\ln x\) by 2 units upwards to obtain the graph of \(f\left( x \right) = \ln x + 2\).

Thus, the graph is obtained.