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what is the domain of this function?
Suppose f(x) = x^2 and g(x) = sqrt(x-1). Then (fog)(x) = f(g(x)) = x - 1. Is the domain of this composition all real numbers? Or do we use the domain of the original function g(x)?
2 Answers
- 4 years ago
for the first one its R
for the second one its x>=1
for the third one it's the same as the second one
- Randy PLv 74 years ago
You include the domain of the original function g(x). The reason is that (f o g)(x) means that this function is NOT simply the operation x - 1, it's consists of these steps:
1. Starting with an x value, calculate y = g(x) = sqrt(x - 1). This only exists if x is in the domain of g(x).
2. Calculate f(y) = y^2. This only exists if y is in the domain of f(x). So you need to actually consider two domains.
The domain of the composition is the domain of those steps, calculating each function individually.
It's the same idea as asking the value of the function x/x. That's obviously equal to 1 everywhere but x = 0? What happens at x = 0? It's not defined. x/x is not the same as 1, because "x/x" says you need to take x and divide it by x as the steps to evaluate it. Which you can't do at x = 0.
