Give the domain of the following functions. Give your answer in interval notation. a) f(x)= 5x-2/x^2-5x+6 and b) square root of 5-2x
a) f(x) = (5x-2)/((x-2)(x-3)) , the brackets are essential.
domain: any real number , except x=2, x=3
b) f(x) = √(5-2x)
we can't take the square root of a negative, so
5-2x ≥ 0
-2x ≥ -5
x ≤ 5/2
To find the domain of a given function, we need to determine the set of all possible values that the input variable (in this case, x) can take. In other words, we are looking for any restrictions or values that would make the function undefined.
a) Let's find the domain of the function f(x) = (5x - 2)/(x^2 - 5x + 6):
First, we need to consider any values of x that would make the denominator equal to zero, as dividing by zero is undefined.
In this case, the denominator is a quadratic expression (x^2 - 5x + 6), so we can check if it can be factored.
The factored form of the denominator is (x - 2)(x - 3).
By setting each factor equal to zero and solving, we can find the values of x that make the denominator zero:
x - 2 = 0 --> x = 2
x - 3 = 0 --> x = 3
So, the function will be undefined when x is equal to 2 or 3.
Therefore, the domain of the function f(x) = (5x - 2)/(x^2 - 5x + 6) is all real numbers except x = 2 and x = 3. In interval notation, this can be written as (-∞, 2) U (2, 3) U (3, +∞).
b) Let's find the domain of the function g(x) = √(5 - 2x):
The square root function (√) is defined for all non-negative real numbers since the square root of a negative number is undefined within the real number system.
For the given function, the expression inside the square root (√(5 - 2x)) must be greater than or equal to zero:
5 - 2x ≥ 0
To solve this inequality, we can isolate x:
-2x ≥ -5
Divide both sides of the inequality by -2, remembering to reverse the inequality sign when dividing by a negative number:
x ≤ 5/2
Therefore, the domain of the function g(x) = √(5 - 2x) is all real numbers x such that x is less than or equal to 5/2. In interval notation, this can be written as (-∞, 5/2].