Question

The function
g
(
x
)
=
(
x
+
4
)
2
−
3
is restricted to the domain
(
−
∞
,
−
4
)
.
Which graph represents the inverse function of
g
(
x
)
?



image.png

image.png

image.png

image.png

Answer 1

Solve y = (x+4)^2 − 3 for x with x < −4:
(x+4)^2 = y+3 so x+4 = −√(y+3) (negative root because x+4 ≤ 0), hence
g⁻¹(y) = −4 − √(y+3).

So the inverse (written as a function of x) is g⁻¹(x) = −4 − √(x+3) with domain x > −3 and range y < −4.

Pick the graph that:
- exists only to the right of the vertical line x = −3,
- stays below the horizontal line y = −4,
- passes through (−2, −5) (since g(−5) = −2),
- and approaches (−3, −4) from below as x → −3⁺.

Equivalently, choose the graph that is the reflection of the left branch of y = (x+4)^2 − 3 across the line y = x. If you can label which of the four images matches those properties, that is the correct choice.