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Determine whether the function f(x) = (x - 7) / (x + 4) has an inverse function. If it does, find the inverse function and state any restrictions on its domain.

User Tyczj
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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.

User McGordon
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