1. Simplify the rational expression: (3x^2 + 5x - 2) / (x^2 - x - 6)
To simplify this expression, first factor the numerator and denominator:
Numerator: 3x^2 + 5x - 2 = (3x - 1)(x + 2)
Denominator: x^2 - x - 6 = (x - 3)(x + 2)
Now simplify the expression by canceling out the common factor in the numerator and denominator:
(3x - 1)(x + 2) / (x - 3)(x + 2) = 3x - 1 / x - 3
So, the simplified expression is 3x - 1 / x - 3.
2. Find the domain of the function: f(x) = 4x / (x^2 - 9)
To find the domain of the function, we need to determine the values of x for which the function is defined. We know that the function is undefined when the denominator is equal to zero. So, we need to find the values of x that make the denominator equal to zero:
x^2 - 9 = 0
(x + 3)(x - 3) = 0
x = 3, x = -3
Therefore, the function is undefined at x = 3 and x = -3. So, the domain of the function is all real numbers except x = 3 and x = -3. In interval notation, the domain is (-∞, -3) U (-3, 3) U (3, ∞).
3. Simplify the expression: (2x^2 - 4x - 6) / (x^2 - 5x + 6)
To simplify this expression, first factor the numerator and denominator:
Numerator: 2x^2 - 4x - 6 = 2(x^2 - 2x - 3) = 2(x - 3)(x + 1)
Denominator: x^2 - 5x + 6 = (x - 2)(x - 3)
Now simplify the expression by canceling out the common factor in the numerator and denominator:
2(x - 3)(x + 1) / (x - 2)(x - 3) = 2(x + 1) / x - 2
So, the simplified expression is 2(x + 1) / x - 2.
4. Find the vertical asymptotes of the function: f(x) = (x^2 + 5x + 6) / (x^2 - 4x)
To find the vertical asymptotes of the function, we need to determine the values of x for which the function is undefined. The function is undefined when the denominator is equal to zero. So, we need to find the values of x that make the denominator equal to zero:
x^2 - 4x = 0
x(x - 4) = 0
x = 0, x = 4
Therefore, the vertical asymptotes of the function are x = 0 and x = 4.
Rational Expressions and Functions Unit Test Part 1
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