ACSEE (Form 6) Functions — practice questions

Explore 4 free Functions questions in ACSEE (Form 6) Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Functions questions and exercises with answers

Showing questions 1–4 of 4.

For $f(x)=\dfrac{x^2-2x-3}{x^2-4}$, the vertical and horizontal asymptotes are:Read the answer and worked solution →For $f(x)=\dfrac{x^2-2x-3}{x^2-4}$, the $x$-intercepts and the $y$-intercept are:Read the answer and worked solution →Tap the asymptotes onlyRead the answer and worked solution →Match a function to its inverseRead the answer and worked solution →

Worked practice questions

For $f(x)=\dfrac{x^2-2x-3}{x^2-4}$, the vertical and horizontal asymptotes are:

$x=\pm 4$ and $y=1$
$x=\pm 2$ and $y=0$
$x=\pm 2$ and $y=1$
$x=\pm 2$ and $y=-3$

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Correct answer: C. $x=\pm 2$ and $y=1$

1. Vertical asymptotes occur where the denominator vanishes: $x^2-4=0$, so $x=\pm 2$. 2. Check the numerator does not vanish there — $x^2-2x-3=(x-3)(x+1)$ is non-zero at $x=\pm2$, so both are genuine asymptotes. 3. Numerator and denominator have equal degree, so the horizontal asymptote is the ratio of leading coefficients. 4. Both leading coefficients are $1$, giving $y=1$. 5. $x=\pm 4$ takes the constant $-4$ as the root instead of solving $x^2=4$; $y=0$ would need the numerator degree to be lower. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 6: Functions_

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For $f(x)=\dfrac{x^2-2x-3}{x^2-4}$, the $x$-intercepts and the $y$-intercept are:

$-1$ and $3$; $y=-\tfrac34$
$1$ and $-3$; $y=\tfrac34$
$-1$ and $3$; $y=\tfrac34$
$-2$ and $2$; $y=\tfrac34$

Show answer and solution

Correct answer: C. $-1$ and $3$; $y=\tfrac34$

1. $x$-intercepts need $f(x)=0$, so set the numerator to zero. 2. $x^2-2x-3=(x-3)(x+1)=0$ gives $x=3$ and $x=-1$. 3. The $y$-intercept is $f(0)=\dfrac{0-0-3}{0-4}=\dfrac{-3}{-4}=\dfrac34$. 4. Both negatives cancel, so the $y$-intercept is positive. 5. $-2$ and $2$ are the asymptotes, not intercepts; the sign-flipped pair mis-factorises the numerator. _Source: NECTA ACSEE 2023 Advanced Mathematics 142/1, Question 6: Functions_

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Tap the asymptotes only

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Questions

Items
1. Text
  x = 2
  Matches the category
  Yes
2. Text
  x = −2
  Matches the category
  Yes
3. Text
  y = 1
  Matches the category
  Yes
4. Text
  x = 3
  Matches the category
  No
5. Text
  x = −1
  Matches the category
  No
6. Text
  y = 3/4
  Matches the category
  No
Category
Asymptote of the curve
Items
1. Text
  x = 3
  Matches the category
  Yes
2. Text
  x = −1
  Matches the category
  Yes
3. Text
  y = 3/4
  Matches the category
  Yes
4. Text
  x = 2
  Matches the category
  No
5. Text
  y = 1
  Matches the category
  No
Category
Intercept of the curve

Instructions

For f(x) = (x²−2x−3)/(x²−4), tap every line that is an asymptote.

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