To find the inverse of the composite function (f o g)(x), denoted as (f o g)⁻¹(x), you typically need to follow these steps:
Steps to Find (f o g)⁻¹(x)
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Find the Composite Function (f o g)(x): First, determine the expression for (f o g)(x), which means f(g(x)). This involves substituting the function g(x) into the function f(x).
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Replace (f o g)(x) with y: Rewrite the composite function as y = (f o g)(x).
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Swap x and y: Interchange x and y in the equation. This gives you x = (f o g)(y).
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Solve for y: Solve the equation obtained in step 3 for y in terms of x. This new expression for y is the inverse function (f o g)⁻¹(x).
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Replace y with (f o g)⁻¹(x): Finally, rewrite your solution, replacing y with the notation (f o g)⁻¹(x).
Example
Let's say f(x) = 2x + 1 and g(x) = x - 3.
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(f o g)(x) = f(g(x)) = f(x - 3) = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5
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y = 2x - 5
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x = 2y - 5
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Solve for y:
x + 5 = 2y
y = (x + 5) / 2 -
(f o g)⁻¹(x) = (x + 5) / 2
Therefore, the inverse of (f o g)(x) is (x + 5) / 2.
Alternative Method: Using Individual Inverses
There's another method which uses the inverses of the individual functions, if you already know them. The following is true:
(f o g)⁻¹(x) = (g⁻¹ o f⁻¹)(x)
This means that the inverse of the composite function (f o g)(x) is equal to the composition of the inverse of f with the inverse of g.
- Find f⁻¹(x): Given f(x), determine its inverse, f⁻¹(x).
- Find g⁻¹(x): Given g(x), determine its inverse, g⁻¹(x).
- Compute (g⁻¹ o f⁻¹)(x): Substitute f⁻¹(x) into g⁻¹(x). This means g⁻¹(f⁻¹(x)).
Using the same example functions:
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f(x) = 2x + 1. To find f⁻¹(x):
- y = 2x + 1
- x = 2y + 1
- x - 1 = 2y
- y = (x - 1) / 2
- Therefore, f⁻¹(x) = (x - 1) / 2
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g(x) = x - 3. To find g⁻¹(x):
- y = x - 3
- x = y - 3
- y = x + 3
- Therefore, g⁻¹(x) = x + 3
Now, let's calculate (g⁻¹ o f⁻¹)(x) = g⁻¹(f⁻¹(x)) = g⁻¹((x-1)/2) = (x-1)/2 + 3 = (x - 1 + 6) / 2 = (x + 5) / 2.
As you can see, this method produces the same result as the first method: (f o g)⁻¹(x) = (x + 5) / 2.
Summary
Finding the inverse of a composite function (f o g)(x) involves finding the composite function, swapping x and y, and solving for y. Alternatively, you can find the inverses of the individual functions, f⁻¹(x) and g⁻¹(x), and then find (g⁻¹ o f⁻¹)(x), which is equivalent to (f o g)⁻¹(x).
