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Discontinuities of Rational Function \( f(x) = \frac{x^2 + 6x}{x^2 + 4x - 12} \)
Mathematics
High School
Question Content
Where will the discontinuities occur in the graph of the rational function \( f(x) = \frac{x^2 + 6x}{x^2 + 4x - 12} \)?
Correct Answer
C
Detailed Solution Steps
1
Factor numerator and denominator: Numerator \( x^2 + 6x = x(x + 6) \); Denominator \( x^2 + 4x - 12 = (x + 6)(x - 2) \)
2
Identify where denominator is zero: \( (x + 6)(x - 2) = 0 \) → \( x = -6 \) or \( x = 2 \)
3
Check numerator at these points: At \( x = -6 \), numerator is 0 (hole); at \( x = 2 \), numerator is 16 (vertical asymptote). Discontinuities at \( x = 2 \) and \( x = -6 \)
Knowledge Points Involved
1
Rational Functions
Functions of the form \( f(x) = \frac{P(x)}{Q(x)} \) (polynomials \( P, Q \)); discontinuities occur where \( Q(x) = 0 \).
2
Removable Discontinuity (Hole)
Occurs when both numerator and denominator are zero (factor cancels), creating a hole in the graph.
3
Vertical Asymptote
Occurs when denominator is zero and numerator is not zero at that point, creating a vertical asymptote.
4
Factoring Polynomials
Breaking down polynomials into factors (e.g., \( x^2 + 4x - 12 = (x + 6)(x - 2) \)) to analyze discontinuities.
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