Question

What are the domain and range of this function?

Answer 1

I don't see the function — could you paste it? Once I have it I can give the exact domain and range and show the steps.

Meanwhile, here are quick rules and methods you can use (and I’ll apply them to your specific function when you send it):

How to find the domain
- Look for values of x that make the expression undefined:
- Denominators: exclude x that make any denominator zero.
- Even roots: require the radicand ≥ 0 for real-valued functions (e.g., √(…) needs inside ≥ 0).
- Logarithms: require the argument > 0 (for real logs).
- Other operations: consider where pieces are undefined (e.g., inverse trig, fractional exponents with noninteger denominators).
- Combine restrictions (intersection). If none apply, domain is all real numbers.

How to find the range
- Algebraic inversion: set y = f(x) and try to solve for x in terms of y; valid y are those for which you can find at least one real x in the domain.
- Use calculus: find minima/maxima or limits to see how outputs behave as x→±∞ or near vertical asymptotes.
- Use known ranges for standard functions (e.g., sin ∈ [−1,1], exp > 0, ln ∈ R).
- For rational functions, find horizontal/oblique asymptotes and solve y = f(x) for x to find excluded y-values.
- For radicals and quadratics, complete the square to find range.

Short examples
- f(x) = √(x−3): domain x ≥ 3; range y ≥ 0.
- f(x) = (x−1)/(x+2): domain x ≠ −2; range y ≠ 1 (solve y = (x−1)/(x+2) for x to see y = 1 has no solution).
- f(x) = x^2: domain (−∞, ∞); range [0, ∞).
- f(x) = ln(x+4): domain x > −4; range (−∞, ∞).
- f(x) = sin x: domain (−∞, ∞); range [−1, 1].

Send the function and I’ll compute the domain and range and show the work.