Determine the inverse of g(x) is a function. If not, restrict the domain of g(x) so the inverse will be a function, then find g^-1(x). g(x)=(x+3)^2 -5

Question

g(x)=(x+3)^2 -5

-5 in not part of the exponent.

Answer 1

The graph of g(x) = (x+3)² - 5 is a parabola opening upward with vertex (-3,-5).

Since the graph fails the Horizontal Line test, the inverse will not be a function.

Restrict the domain to [-3, ∞), The restricted graph (the right half of the parabola) does satisfy the Horizontal Line Test. So, the inverse of the restricted function is also a function.

Answer 2

y = (x+3)² - 5

To find the inverse, exchange the roles of x and y and solve for y.

x = (y+3)² - 5

(y+3)² = x+5

y+3 = √(x+5)

y = -3 + √(x+5)

Now I recommend that you graph these 2 equations along with y=x.,

If they are indeed inverse, the graphs will be symmetric with respect to y = x.

Of course, g^-1(g(x)) should = x and you should try that, too!