Reading f(x)
Function notation writes the output of the function f at input x as f(x), read “f of x.” It is not multiplication — f(x) does not mean f times x. The name f labels the rule; whatever sits in the parentheses is the input you hand it. So if f(x) = 3x − 1, then f(2) means “run the rule on 2”: replace every x by 2.
f(x) = 3x - 1 f(2) = 3(2) - 1 = 6 - 1 = 5 f(-4) = 3(-4) - 1 = -12 - 1 = -13 f(0) = 3(0) - 1 = 0 - 1 = -1 f(a+1)= 3(a+1) - 1 = 3a + 3 - 1 = 3a + 2
Domain: which inputs are allowed
The domain is the set of all inputs the rule can legally accept. For a plain polynomial like 3x − 1, every real number works, so the domain is all reals. Trouble appears in only two common places: you cannot divide by zero, and you cannot take an even root of a negative.
- For g(x) = 1/(x − 5), set the denominator to zero: x − 5 = 0 gives x = 5. That is an excluded value, so the domain is all reals except 5.
- For h(x) = sqrt(x − 2), demand the radicand be ≥ 0: x − 2 ≥ 0 gives x ≥ 2. The domain is x ≥ 2.
- Write the answer in interval notation when you can: x ≥ 2 becomes [2, ∞).
Range: which outputs come out
The range is the set of outputs the function actually produces as x sweeps over the whole domain. It is often harder to find than the domain because you have to think about what y-values are reachable. A good first move is to ask which values are *impossible*.
f(x) = x^2 x^2 is never negative, and every value >= 0 is hit. Domain: all reals. Range: y >= 0, i.e. [0, infinity). f(x) = sqrt(x - 2) Domain: x >= 2. A square root is never negative -> Range: y >= 0, i.e. [0, infinity).