Question

Evaluate each function over the set of real numbers.

es001-1.jpg

Answer 1

I can’t see the image you attached. Could you either paste the functions here as text or describe them?

If you meant “evaluate each function over the set of real numbers” as in “find the domain and/or range over R,” here is how to proceed and some common examples:

How to determine domain (set of real x where the function is defined)
- Polynomials (e.g. x^2 + 3x − 5): domain = all real numbers, R.
- Rational functions (e.g. (x+1)/(x−3)): exclude x that make the denominator 0 (here x ≠ 3).
- Even roots (square root, fourth root): require the radicand ≥ 0 (e.g. √(x−1) ⇒ x ≥ 1).
- Logarithms: argument must be > 0 (e.g. ln(x+2) ⇒ x > −2).
- Absolute value: defined for all real x.
- Trig functions: sin, cos defined for all real x; tan undefined where cos x = 0 (x = π/2 + kπ).
- Exponential functions e^x defined for all real x.

How to find the range (set of possible y values)
- For y = x^2, range y ≥ 0.
- For y = 1/(x−3), as x varies over its domain the function can take all real values except 0 (range R\{0}).
- For y = √(x−1), range y ≥ 0.
- For y = |x| − 2, range y ≥ −2.
- For y = e^x, range y > 0.
- For y = ln(x+2), range all real numbers.

Examples:
- f(x) = x^2: domain = R, range = [0, ∞).
- g(x) = 1/(x−3): domain = R\{3}, range = R\{0}.
- h(x) = √(x−1): domain = [1, ∞), range = [0, ∞).
- p(x) = ln(x+2): domain = (−2, ∞), range = R.

If you paste the exact functions from your problem I’ll compute the domain and range (or evaluate them) for each one.