Find the domain and range of the following functions

Question

f(x)=√(1-x^2)
f(x)=(x-4)/(x^2-2x-15)
f(x)=In(x-8)

Answer 1

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range of a function is the set of all possible output values (y-values) that the function can produce.

1. f(x) = √(1 - x^2):
- The square root function is only defined for non-negative real numbers.
- The expression 1 - x^2 must be non-negative for the function to be defined.
- This means that 1 - x^2 ≥ 0.
- Solving the inequality, we get x^2 ≤ 1.
- Taking the square root of both sides (since the radical function is positive), we get -1 ≤ x ≤ 1.
- Therefore, the domain of the function is [-1, 1].
- The range of the function is [0, 1] since for any input value x in the domain, the output value f(x) will be a non-negative value less than or equal to 1.

Domain: [-1, 1]
Range: [0, 1]

2. f(x) = (x - 4)/(x^2 - 2x - 15):
- To find the domain of this rational function, we need to determine where it is defined and exclude any x-values that would result in a denominator of 0.
- The denominator x^2 - 2x - 15 can be factored as (x - 5)(x + 3).
- The function is not defined when the denominator is equal to 0, so we set x - 5 = 0 and x + 3 = 0.
- Solving these equations, we find x = 5 and x = -3.
- Therefore, the function is not defined at x = 5 and x = -3, so the domain of the function is all real numbers except 5 and -3.
- The range of the function will depend on the behavior of the function as x approaches ±∞.
- As x approaches ±∞, the function approaches 0.
- Therefore, the range of the function is all real numbers except for 0.

Domain: (-∞, -3) U (-3, 5) U (5, ∞)
Range: R - {0} (where R represents all real numbers)

3. f(x) = ln(x - 8):
- The natural logarithm function is defined only for positive real numbers.
- Therefore, the expression x - 8 must be greater than 0.
- Solving the inequality, we get x > 8.
- Hence, the domain of the function is (8, ∞).
- The range of the natural logarithm function is all real numbers.

Domain: (8, ∞)
Range: (-∞, ∞)