Question

please show work

1.Let g(x) = log3(x +3)-1 . d. (3 pts) f(8)-3, the corresponding point on the graph of f(x)is.H The transformed point on the graph of g(x) is . e. (2 pts) Find the domain and the range. Write in interval notation. 1d. point on f(x): point on g(x): f. (1 pt) What is the vertical asymptote? That is, as x→ 1e. D: R: 1f. 8. (5 pts) Find the equation of the inverse, g(x). 1g.

Answer 1

As the function f(x) isn't mentioned here nor its graph, so, I can't help you with part 'd' of the question.

I'm assuming that the parts 'e' and 'f' are related to g(x) only, so,

(e) g(x) = log₂(x + 3) - 1

And log can have only positive values, so, x + 3 must be greater than 0.

Hence, Domain = (-3, infinity)

For Range,

Value of log x goes from -infinity to infinity, so, Range of g(x) would also be (-infinity, infinity).

(f) The function is undefined at x = -3, or it becomes -infinity there.

So, the vertical asymptote is at x = -3.

(g) To find the inverse of g(x),

We need to interchange 'x' and 'y' in the given equation of g(x) and solve it for y,

x = log₂(y + 3) - 1

or, y + 3 = 2^{x + 1}

or, y = 2^{x + 1} - 3

Hence, g^-1(x)= 2^{x + 1} - 3.