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Which graph best represents the curve $y = \frac{1}{\sqrt{x^2 - 4}}$? - HSC - SSCE Mathematics Extension 2 - Question 4 - 2018 - Paper 1
Question 4
Which graph best represents the curve $y = \frac{1}{\sqrt{x^2 - 4}}$?
Worked Solution & Example Answer:Which graph best represents the curve $y = \frac{1}{\sqrt{x^2 - 4}}$? - HSC - SSCE Mathematics Extension 2 - Question 4 - 2018 - Paper 1
Step 1
Identify the Function Behavior
Answer
The given function is defined as . For the square root to be real and positive, the expression in the square root must be greater than zero, which gives us the inequality:
$$ \text{or} \\ $$x^2 > 4. \\This simplifies to ( x > 2 ) or ( x < -2 ). Therefore, the function has vertical asymptotes at ( x = 2 ) and ( x = -2 ).
Step 2
Evaluate Asymptotic Behavior
Answer
As the function approaches these asymptotes, the value of ( y ) will increase without bound (tending to ( +\infty )). This means as we approach ( x = 2 ) from the left, ( y \to +\infty ) and as we approach ( x = 2 ) from the right, ( y \to 0 ). Similarly, we will have ( y \to +\infty ) as we approach ( x = -2 ) from the right and ( y \to 0 ) from the left.
Step 3