The domain of the function f(x) is open square bracket, 20, comma, infinity, right bracket[20,∞) and the range is open square bracket, minus, 16, comma, minus, 2, close square bracket[−16,−2]. Using interval notation, find the domain and range of g, of, x, equals, f, of, 5, xg(x)=f(5x).
Question
The domain of the function f(x) is open square bracket, 20, comma, infinity, right bracket[20,∞) and the range is open square bracket, minus, 16, comma, minus, 2, close square bracket[−16,−2]. Using interval notation, find the domain and range of g, of, x, equals, f, of, 5, xg(x)=f(5x).
Solution
The domain and range of a function can be affected when the function is transformed. In this case, the function g(x) = f(5x) is a horizontal compression of the function f(x) by a factor of 1/5.
Step 1: Determine the new domain The domain of f(x) is [20,∞). Since g(x) = f(5x), we need to divide the domain of f(x) by 5 to get the domain of g(x). So, the domain of g(x) is [20/5, ∞/5) = [4, ∞).
Step 2: Determine the new range The range of f(x) is [−16,−2]. The range of g(x) = f(5x) is the same as the range of f(x) because a horizontal compression or stretch does not affect the range of the function. So, the range of g(x) is [−16,−2].
So, in interval notation, the domain of g(x) is [4, ∞) and the range is [−16,−2].
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