1. Find the domain and range of each function.

Question

i) f(x) = 1 - √𝑥 ii) F(x) = √5𝑥 + 10 iii) g(x) = √𝑥^2 − 3𝑥 iv) G(t) = 2/𝑡^2−16 v) f(x) = 1 − 2𝑥 − 𝑥^2 vi) F (t) = 𝑡/|𝑡| vii) g(x) = √|𝑥| viii) f(x) = 5 – 2x

please answer all questions in details . thank you

Answer 1

A few important things to remember about certain functions and domain:

1) just plain polynomials (not in a rational function) - the domain is always "all real numbers".

2) Rational functions (usually consisting of 2 polynomials being divided) - the domain cannot allow the denominator to be equal to zero. So to find the values that are NOT in the domain, set the denominator equal to zero and solve. The domain is everything else.

3) Square root or radical functions - for square root or radicals with an even index (4th root, 6th root . . .) - what's under the radical MUST be greater than or equal to zero. So to find the domain, set what's under the radical > zero and solve. The result is the domain.

Example: Problem i) fits into item 3) above. "x" is what's under the radical so x > 0 is the domain. The range is what results. y < 1

Example: Problem iv) fits into item 2) above. t² - 16 = 0 or (t - 4)(t + 4) = 0 when t = 4 or t = -4 so the domain is everything else (-∞, -4) U (-4, 4) U (4, ∞). The range is y ≠ 0 so (-∞, 0) U (0, ∞)

Problem vii) has an absolute value under the radical making the results under the radical ALWAYS positive so, in that case, the domain is all real numbers. But the range will just be y > 0.

You should be able to determine the others with the information provided.