Question

graph f and g in the same viewing rectangle. then describe the relationship of the graph of g to the graph of f. f(x)=log x, g(x)=log (x + 2)-1

Explanation:

Step1: Recall transformation rules

For a function \(y = f(x + h)+k\), if \(h>0\), the graph of \(y = f(x)\) is shifted \(h\) units to the left; if \(k < 0\), the graph of \(y = f(x)\) is shifted \(|k|\) units down.

For \(f(x)=\log x\) and \(g(x)=\log(x + 2)-1\), here \(h = 2\) and \(k=-1\).

Step2: Analyze the transformation

For the function \(g(x)=\log(x + 2)-1\) compared to \(f(x)=\log x\):
The \(x\) in \(f(x)\) is replaced by \(x + 2\). According to the rule \(y=f(x + h)\) (where \(h = 2\)), the graph of \(y = f(x)\) is shifted \(2\) units to the left.
Then, we have a subtraction of \(1\) from the function \(\log(x + 2)\). According to the rule \(y=f(x)+k\) (where \(k=-1\)), the graph of \(y=\log(x + 2)\) is shifted \(1\) unit down.

Answer:

The graph of \(g(x)\) is the graph of \(f(x)\) shifted \(2\) units to the left and \(1\) unit down.