f⁻¹(x)= 2x – 10
f(x)= ½(x) + 5
The given equation is: f⁻¹(x) = 2x – 10
This equation is representing the inverse function of f(x).
Here’s how to find the original function f(x):
Step 1: Replace f(x) with y. This will give you: y = 2x – 10
Step 2: Interchange x and y. This means that we switch x for y and y for x: x = 2y – 10
Step 3: Solve for y:
Add 10 to both sides of the equation: x + 10 = 2y
Divide both sides by 2: (x + 10) / 2 = y
Thus, we found the original function f(x): f(x) = (x + 10) / 2.
To verify that f(x) and f⁻¹(x) are indeed inverse functions of each other, we need to check whether f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Substituting f⁻¹(x) into f(x), we get:
f(f⁻¹(x)) = f((2x – 10))
= [(2x – 10) + 10] / 2
= 2x / 2
= x
This shows that f(x) and f⁻¹(x) are indeed inverse functions of each other.
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