Let f (x) = √x − 1 and g(x) = 1x − 2 .(i) State the domains for f and g, and the range of f .
Question
Let f (x) = √x − 1 and g(x) = 1x − 2 .(i) State the domains for f and g, and the range of f .
Solution
(i)
The domain of a function is the set of all possible input values (x-values) which will output real numbers.
For the function f(x) = √x - 1, the input x must be greater than or equal to 1. This is because the square root of a negative number is not a real number. So, the domain of f is [1, ∞).
For the function g(x) = 1/(x - 2), the denominator cannot be zero because division by zero is undefined. Therefore, x cannot be equal to 2. So, the domain of g is (-∞, 2) U (2, ∞).
The range of a function is the set of all possible output values (y-values). For the function f(x) = √x - 1, the smallest value is 0 (which occurs when x = 1) and it can increase without bound as x increases. Therefore, the range of f is [0, ∞).
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