Question

f(x) has domain
, and range
.

Answer 1

It looks like part of your question is missing — you wrote "f(x) has domain , and range ." — please paste the function f(x) or a graph so I can find the domain and range.

While you paste it, here is a quick guide on how to determine domain and range and some common examples:

How to find the domain
- Start with all real numbers, then exclude values that make the expression undefined:
- Denominators: exclude x where denominator = 0 (rational functions).
- Even roots: require radicand ≥ 0 (square roots, 4th roots, etc.).
- Logarithms: require argument > 0.
- Trig: exclude points where denominator trig function = 0 (tan, sec, etc.).
- Piecewise: take union of domains of pieces.
- Polynomials: domain = all real numbers.
- For parametric/implicit definitions, domain = allowed parameter values or x-values that produce real outputs.

How to find the range
- Solve y = f(x) for x in terms of y (if possible), then determine allowable y values.
- Use monotonicity/inverse: if f is one-to-one, the range is the domain of f^{-1}.
- Use calculus: find extrema and limits as x → ±∞, and inspect asymptotes and discontinuities.
- For graphs: project the graph onto the y-axis (all y-values attained).
- For piecewise: take union of ranges of pieces.

Examples
- f(x) = 1/(x − 2) → domain: (-∞, 2) ∪ (2, ∞); range: (-∞, 0) ∪ (0, ∞) (y ≠ 0).
- f(x) = √(x − 3) → domain: [3, ∞); range: [0, ∞).
- f(x) = x^2 → domain: (−∞, ∞); range: [0, ∞).
- f(x) = sin x → domain: (−∞, ∞); range: [−1, 1].

If you post the specific f(x) (or a picture of its graph), I’ll compute the exact domain and range and show the steps.