Question

question 8 of 25
which of these could be the graph of ( f(x)=ln x + 2 )?

a. graph a
b. graph b
c. graph c
d. graph d

Explanation:

Step1: Recall the parent function

The parent function of \(F(x)=\ln x + 2\) is \(y = \ln x\). The domain of \(y=\ln x\) is \(x>0\), and it passes through the point \((1,0)\).

Step2: Analyze the transformation

For the function \(y=\ln x+2\), it is a vertical shift of the function \(y = \ln x\) by 2 units up. So it should pass through the point \((1,2)\).

Step3: Check the graphs

• Graph A: Does not pass through \((1,2)\) (since when \(x = 1\), \(y\) is not 2).
• Graph B: Passes through \((1,2)\) (when \(x = 1\), \(y=\ln(1)+2=0 + 2=2\)).
• Graph C: Domain seems incorrect (if we assume standard axes, the \(x\) - values for which the function is defined).
• Graph D: The function \(y=\ln x+2\) is an increasing function (since the derivative of \(y=\ln x+2\) is \(y'=\frac{1}{x}>0\) for \(x > 0\)), and Graph D is a decreasing function.

Answer:

B. Graph B