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Let f(x) = 2x^2 + x - 3 and g(x) = x - 1. Perform the function operation and then find the domain.
f(x)/g(x)
Note: x^2 is x squared
To perform the function operation f(x)/g(x), we need to divide f(x) by g(x). Let's substitute the given functions into the expression:
f(x)/g(x) = (2x^2 + x - 3)/(x - 1)
To simplify this expression, we can use polynomial long division. Here's how to do it step by step:
Step 1: Divide the first term of the dividend (2x^2) by the first term of the divisor (x). The result is 2x.
2x + _____
x - 1
Step 2: Multiply the divisor (x - 1) by the quotient (2x), and subtract it from the dividend.
2x(x - 1) = 2x^2 - 2x
(2x^2 + x - 3) - (2x^2 - 2x) = 3x - 3
Step 3: Bring down the next term from the dividend. In this case, it's the constant term -3.
2x + _____
x - 1
- (3x - 3)
Step 4: Repeat steps 1 to 3 until you have subtracted all the terms from the dividend.
2x + 3
At this point, we have the quotient 2x + 3, or (2x + 3)/(x - 1). Now, let's find the domain of this function.
The domain of a function is the set of all possible values that x can take. To determine the domain, we need to consider any restrictions that could make the function undefined.
In this case, the denominator of our function is x - 1. To keep the function defined, we must exclude any values of x that would make the denominator equal to zero. Therefore, x cannot be equal to 1.
Hence, the domain of f(x)/g(x) is all real numbers except 1 or, in interval notation, (-∞,1) ∪ (1,∞).