Answer:
g(x) = log(x+1) +4
Explanation:
The transformation ...
g(x) = f(x -h) +k
translates the graph of f(x) to the right h units and up k units.
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Your graph shows g(x) is a translation of f(x) to the left 1 unit and up 4 units. That means the graph of g(x) can be described by ...
g(x) = f(x -(-1)) +4
g(x) = log(x+1) +4
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Additional comment
The vertical asymptote of f(x) at x=0 provides an identifiable feature for assessing the amount of horizontal shift. The vertical asymptote at x=-1 for g(x) means there was a left shift of 1 unit.
One unit to the right of the vertical asymptote, the graph of f(x) has an x-intercept (y=0). On the graph of g(x), one unit to the right of the vertical asymptote, the graph shows a value of y=4. This suggests an upward shift of 4 units.
We observe that the graph of g(x) generally matches the shape of the graph of f(x), so there is likely no horizontal or vertical scaling. A horizontal expansion factor of 'a' and a vertical expansion factor of 'b' would modify the function to ...
g(x) = b·f(x/a)