Final answer:
The function f(x) = (x - 7) / (x + 4) has an inverse; its inverse function is f⁻¹(x) = (-7 - 4x) / (x - 1) and there's a restriction on its domain where x cannot equal 1 to avoid undefined division.
Step-by-step explanation:
To determine whether the function f(x) = (x - 7) / (x + 4) has an inverse, we first need to check if the function is one-to-one. This can be done by using the horizontal line test or by showing that for any two points x1 and x2, if x1 ≠ x2, then f(x1) ≠ f(x2). Since the function f(x) is a rational function without repeated x-values for the y-values, we can say that the function does have an inverse.
To find the inverse function, we first replace f(x) with y:
y = (x - 7) / (x + 4).
Next, we switch x and y and solve for y:
x = (y - 7) / (y + 4).
To solve for y, we cross-multiply:
x(y + 4) = y - 7.
Now we expand and collect all the y terms on one side:
xy + 4x = y - 7.
Rearrange the terms:
xy - y = -7 - 4x.
Factor out y:
y(x - 1) = -7 - 4x.
Divide by (x - 1) to isolate y:
y = (-7 - 4x) / (x - 1).
The inverse function is f-1(x) = (-7 - 4x) / (x - 1). There are restrictions on the domain of the inverse function, as x cannot be equal to 1 because it would make the denominator of the inverse equal to zero, which is undefined.