give the domain, range, and zeros of:
f(x)=absoulte value of x-3.
f(x)=x^2-4x-8
f(x)=(x+1)(x+1)(x+1).
f(x)=4
f(x)=-'9x-1)^2+16
y=squareroot x.
i don't understand how i konw what the range, domain, and zeros are...please give some answers so i get it
To find the domain, range, and zeros of a function, it is important to know what each term means:
1. Domain: The domain of a function is the set of all possible input values (x-values) for which the function is defined.
2. Range: The range of a function is the set of all possible output values (y-values) that the function can produce.
3. Zeros: The zeros of a function, also known as the x-intercepts or roots, are the values of x for which the function equals zero.
Let's apply these concepts to the given functions one by one:
1. f(x) = |x - 3|:
- Domain: This is an absolute value function, meaning it is defined for all values of x. Therefore, the domain is all real numbers (-∞, ∞).
- Range: The absolute value function ensures that the output (y-value) is always non-negative. Thus, the range is [0, ∞).
- Zeros: To find the zeros, set f(x) = 0 and solve for x: |x - 3| = 0 ➝ x - 3 = 0 ➝ x = 3. So, the zero is x = 3.
2. f(x) = x^2 - 4x - 8:
- Domain: In this case, the function is a polynomial, which is defined for all real numbers. Hence, the domain is (-∞, ∞).
- Range: Since this is a quadratic function opening upward (positive leading coefficient), the range is [min, ∞). You can find the minimum value by finding the vertex or using calculus.
- Zeros: To find the zeros, set f(x) = 0 and solve for x using factoring, completing the square, or the quadratic formula.
3. f(x) = (x + 1)(x + 1)(x + 1):
- Domain: Similar to the previous case, this polynomial function is defined for all real numbers.
- Range: In this case, since it is also a polynomial, the range is (-∞, ∞).
- Zeros: To find zeros, set f(x) = 0 and solve for x using any suitable method.
4. f(x) = 4:
- Domain: This is a constant function, meaning it is defined for all real numbers.
- Range: The output is always 4 for any input. So, the range is just {4}.
- Zeros: Since f(x) = 4 for all x, it has no zeros.
5. f(x) = -(9x - 1)^2 + 16:
- Domain: Again, this is a polynomial function, so it is defined for all real numbers.
- Range: The given function is a quadratic function with a negative leading coefficient, so its range is (−∞, max]. You can find the maximum value using calculus or by completing the square.
- Zeros: To find the zeros, set f(x) = 0 and solve for x.
6. y = √x (Square Root Function):
- Domain: For the square root function, the input (x) must be non-negative since you cannot take the square root of a negative number. Therefore, the domain is [0, ∞).
- Range: The output of the square root function is always non-negative, so the range is [0, ∞).
- Zeros: To find the zeros, set y = 0 and solve for x.
Please note that for more complicated functions, finding the range and zeros may require additional steps or techniques.