Find the domain and range of the following functions
f(x)=√(1-x^2)
f(x)=(x-4)/(x^2-2x-15)
f(x)=In(x-8)
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: (-∞, ∞)
To find the domain and range of the given functions, we will analyze each function separately.
1. f(x) = √(1-x^2):
Domain: The square root function is defined for any non-negative real number inside the square root. Therefore, the radicand (1-x^2) must be greater than or equal to zero:
1 - x^2 ≥ 0
To solve the inequality, we can factor it as:
(1-x)(1+x) ≥ 0
The critical points are x = -1 and x = 1.
Considering the sign of the inequality in different intervals:
For x < -1, both (1-x) and (1+x) will be negative, making the inequality false.
For -1 < x < 1, (1-x) will be negative, and (1+x) will be positive, making the inequality true.
For x > 1, both (1-x) and (1+x) will be positive, making the inequality true.
Therefore, the domain of f(x) = √(1-x^2) is the interval (-1, 1).
Range: The square root of a non-negative number is always non-negative. So, the range of f(x) is [0, ∞).
2. f(x) = (x-4)/(x^2-2x-15):
Domain: The function has a rational expression. To determine the domain, we need to find any restrictions on x.
The denominator (x^2 - 2x - 15) must not equal zero to avoid division by zero. So, we solve:
x^2 - 2x - 15 ≠ 0
Factoring the quadratic equation:
(x - 5)(x + 3) ≠ 0
The critical points are x = 5 and x = -3. Thus, the domain of f(x) is the set of all real numbers except x = 5 and x = -3.
Range: To find the range, we can take the limit as x approaches infinity and negative infinity. As the denominator grows larger, the entire fraction approaches zero. Hence, the range of f(x) is all real numbers except zero.
3. f(x) = ln(x-8):
Domain: The logarithmic function is defined only for positive values. Thus, the expression inside the logarithm (x-8) must be greater than zero:
x - 8 > 0
Solving the inequality:
x > 8
Therefore, the domain of f(x) = ln(x-8) is x > 8.
Range: The range of the natural logarithm function is all real numbers.
In summary:
1. f(x) = √(1-x^2):
Domain: (-1, 1)
Range: [0, ∞)
2. f(x) = (x-4)/(x^2-2x-15):
Domain: All real numbers except x = 5 and x = -3
Range: All real numbers except zero
3. f(x) = ln(x-8):
Domain: x > 8
Range: All real numbers