Domain of
a) x^20
b) 1/(x-3)
c)1/(4x^2-21x-18)
d) sqrt(4x^2-21x-18)
e) 1/ sqrt(4x^2-21x-18)
Domain
a. x^20. All real values of x.
b. 1/(x-3).
3 is not in the domain, because it gives a denominator of 0. So the domain is all real values of x except 3:
3 > x >3.
c. 1 / (4x^2-21x-18) =
1 / (x-6)(4x+3) .
Domain: All real values of x except 6:
6 > x > 6.
d. sqrt(4x^2-21x-18) =
sqrt((x-6)(4x+3)).
Domain: All real values of x that are
=> 6. Values below 6 will result in a
negative value under the radical.
Domain: X => 6.
e. 1 / sqrt(4x^2-21x-18) =
1 / sqrt((x-6)(4x+3)).
6 is not in the domain,because it gives a denominator of 0. Values below 6 results in a negative value under the radical. Therefore, all values greater than 6 are in the domain:
Domain: X > 6.
1. sqrt x^2-5x-14
2. cube root x-6/sqrt x^2-x-30
3. y= ln2x-12
To find the domain of a function, we need to determine the values for which the function is defined. Generally, the domain consists of all the real numbers that make the function meaningful and avoid any undefined operations. Let's analyze each function separately:
a) x^20:
The domain of this function is all real numbers because any real number raised to the 20th power is defined.
b) 1/(x-3):
Since division by zero is undefined, we need to find the values of x that make the denominator (x-3) equal to zero. So, x-3 = 0, which implies x = 3. Hence, the domain of this function is all real numbers except x = 3.
c) 1/(4x^2-21x-18):
Similar to the previous function, we need to find the values of x that make the denominator (4x^2-21x-18) equal to zero. We can factorize the denominator as (4x+3)(x-6). Setting each factor equal to zero, we have 4x+3 = 0, which gives x = -3/4, and x-6 = 0, which gives x = 6. Therefore, the domain of this function is all real numbers except x = -3/4 and x = 6.
d) sqrt(4x^2-21x-18):
The domain of this function is determined by the values that avoid taking the square root of a negative number. Inside the square root, we have 4x^2-21x-18. To find the values that result in a non-negative number, we need to determine the roots of this quadratic equation. If the quadratic is factorable, we can find the values of x when the quadratic expression is equal to zero. If the quadratic is not factorable, we can use the discriminant (∆ = b^2 - 4ac) to determine the nature of the roots.
e) 1/sqrt(4x^2-21x-18):
Similar to the previous function, we need to consider the values that avoid taking the square root of a negative number. Inside the square root, we have 4x^2-21x-18. The condition for the value inside the square root to be non-negative still applies. The domain will exclude the values that make the denominator zero since division by zero is undefined.
To summarize:
a) Domain: All real numbers
b) Domain: All real numbers except x = 3
c) Domain: All real numbers except x = -3/4 and x = 6
d) Domain: Determined by the roots of the quadratic 4x^2-21x-18
e) Domain: Determined by the roots of the quadratic 4x^2-21x-18 and excluding x-values that make the denominator zero.