Question

What values are excluded from the domain and range of the function mc017-1.jpg?
domain: mc017-2.jpg

range: mc017-3.jpg
domain: mc017-4.jpg
range: mc017-5.jpg
domain: mc017-6.jpg
range: mc017-7.jpg
domain: mc017-8.jpg
range: mc017-9.jpg

Answer 1

I can’t see the images you referenced. Please either paste the function expressions or upload the images.

Meanwhile, here’s a quick guide you can use to find which values are excluded from domain and range, plus a few short examples.

How to find excluded domain values
- Rational functions: exclude any x that makes the denominator 0. If a factor cancels (hole), that x is still excluded from the domain.
- Even roots (√, 4th root, …): require the radicand ≥ 0; exclude x that make it negative.
- Logarithms: argument must be > 0; exclude x that make it ≤ 0.
- Other expressions: exclude x that make expressions undefined (division by zero, negative inside even root, etc.).

How to find excluded range values
- Solve y = f(x) for x in terms of y; find any y for which no x satisfies the equation (often because solving produces a denominator 0 or contradiction). Those y are excluded from the range.
- For rational (linear/linear) functions f(x) = (ax+b)/(cx+d), the value y = a/c (horizontal asymptote) is typically excluded if the equation f(x) = a/c has no solution — check by solving for x.
- For functions with a hole: if the simplified expression attains value L at the hole x0 but the function is undefined there, L is excluded from the range unless some other x gives that value.
- For sqrt: range is typically y ≥ 0 (or shifted), so exclude values less than that lower bound.
- For ln: range is all real numbers (no exclusions).

Quick examples
- f(x) = (x^2 − 1)/(x − 1). Factor: (x−1)(x+1)/(x−1) → simplifies to x+1 but x = 1 is a hole. Domain: all real x except x = 1. Range: all real y except y = 2 (because the simplified rule gives f(1)=2 but the function is undefined there; no x gives y = 2).
- f(x) = 1/(x−3). Domain: x ≠ 3. Range: y ≠ 0 (there is no x with 1/(x−3) = 0).
- f(x) = sqrt(x−2). Domain: x ≥ 2. Range: y ≥ 0.
- f(x) = ln(x+1). Domain: x > −1. Range: all real numbers.

If you paste the five functions (or upload the images), I’ll give the excluded domain and range values for each one.