Grade 12 Maths Paper 1 Revision Document Request
1. Functions and Inverses
In Grade 12, we expand our knowledge of functions to include their inverses. The
inverse function, denoted as 𝑓 −1 , essentially “undoes” what the original function 𝑓 did.
Graphically, the inverse is a reflection of the original function across the line 𝑦 = 𝑥.
• Horizontal Line Test: A function has an inverse that is also a function only if any
horizontal line passes through the graph of 𝑓 no more than once.
• Parabolas: Since 𝑓(𝑥) = 𝑎𝑥 2 fails the horizontal line test, we must restrict the
domain (usually to 𝑥 ≥ 0 or 𝑥 ≤ 0) to make its inverse a function.
, Problem 1 Given 𝑓(𝑥) = 3𝑥 2 for 𝑥 ≤ 0:
• 1.1. Write down the equation of 𝑓 −1 in the form 𝑦 =. ..
• 1.2. State the domain and range of 𝑓 −1 .
Solution
• 1.1. To find the inverse, swap 𝑥 and 𝑦:
𝑥 = 3𝑦 2
𝑥
𝑦2 =
3
𝑥
𝑦 = ±√
3
Since the original domain was 𝑥 ≤ 0, the range of the inverse must be 𝑦 ≤ 0.
Therefore:
𝑥
𝑦 = −√
3
• 1.2. The range of 𝑓 becomes the domain of 𝑓 −1 . For 𝑓(𝑥) = 3𝑥 2 , if 𝑥 ≤ 0, then
𝑦 ≥ 0. Domain of 𝑓 −1 : 𝑥 ≥ 0 Range of 𝑓 −1 : 𝑦 ≤ 0
2. Financial Mathematics
This section focuses on Annuities, which are sequences of equal payments made at
regular intervals.
• Future Value Annuity: Used for savings or “sinking funds.”
𝑥[(1 + 𝑖)𝑛 − 1]
𝐹=
𝑖
• Present Value Annuity: Used for loans or withdrawals from a retirement fund.
𝑥[1 − (1 + 𝑖)−𝑛 ]
𝑃=
𝑖
• Variables: 𝑥 is the regular payment, 𝑖 is the interest rate per period, and 𝑛 is the
total number of payments.
Problem 2 Teboho takes out a loan of R 250 000 to start a business. The bank charges
interest at 12% per annum compounded monthly. He repays the loan with equal monthly
installments over 5 years, starting one month after the loan was granted. Calculate his
monthly installment.
1. Functions and Inverses
In Grade 12, we expand our knowledge of functions to include their inverses. The
inverse function, denoted as 𝑓 −1 , essentially “undoes” what the original function 𝑓 did.
Graphically, the inverse is a reflection of the original function across the line 𝑦 = 𝑥.
• Horizontal Line Test: A function has an inverse that is also a function only if any
horizontal line passes through the graph of 𝑓 no more than once.
• Parabolas: Since 𝑓(𝑥) = 𝑎𝑥 2 fails the horizontal line test, we must restrict the
domain (usually to 𝑥 ≥ 0 or 𝑥 ≤ 0) to make its inverse a function.
, Problem 1 Given 𝑓(𝑥) = 3𝑥 2 for 𝑥 ≤ 0:
• 1.1. Write down the equation of 𝑓 −1 in the form 𝑦 =. ..
• 1.2. State the domain and range of 𝑓 −1 .
Solution
• 1.1. To find the inverse, swap 𝑥 and 𝑦:
𝑥 = 3𝑦 2
𝑥
𝑦2 =
3
𝑥
𝑦 = ±√
3
Since the original domain was 𝑥 ≤ 0, the range of the inverse must be 𝑦 ≤ 0.
Therefore:
𝑥
𝑦 = −√
3
• 1.2. The range of 𝑓 becomes the domain of 𝑓 −1 . For 𝑓(𝑥) = 3𝑥 2 , if 𝑥 ≤ 0, then
𝑦 ≥ 0. Domain of 𝑓 −1 : 𝑥 ≥ 0 Range of 𝑓 −1 : 𝑦 ≤ 0
2. Financial Mathematics
This section focuses on Annuities, which are sequences of equal payments made at
regular intervals.
• Future Value Annuity: Used for savings or “sinking funds.”
𝑥[(1 + 𝑖)𝑛 − 1]
𝐹=
𝑖
• Present Value Annuity: Used for loans or withdrawals from a retirement fund.
𝑥[1 − (1 + 𝑖)−𝑛 ]
𝑃=
𝑖
• Variables: 𝑥 is the regular payment, 𝑖 is the interest rate per period, and 𝑛 is the
total number of payments.
Problem 2 Teboho takes out a loan of R 250 000 to start a business. The bank charges
interest at 12% per annum compounded monthly. He repays the loan with equal monthly
installments over 5 years, starting one month after the loan was granted. Calculate his
monthly installment.
