Mathematics (Syllabus D) 4024/22 — May/June 2011

Walkthrough

When adding or subtracting algebraic fractions, the key step is finding a common denominator. The denominators here are $2x$ and $5x$. Since 2 and 5 are prime to each other, their least common multiple is $2 \times 5 = 10$. Both terms contain $x$, so the common denominator for the variables is just $x$. Combining these gives $10x$.

We multiply the numerator and denominator of the first fraction by 5 to get the denominator from $2x$ to $10x$. Similarly, we multiply the numerator and denominator of the second fraction by 2 to get the denominator from $5x$ to $10x$. Once the denominators are the same, we simply subtract the top numbers (numerators): $5 - 4 = 1$. The result is $\frac{1}{10x}$. This is already in its simplest form because 1 and $10x$ share no common factors.

Key Takeaways

• The lowest common multiple (LCM) of $ax$ and $bx$ is $(ab)x$ if $a$ and $b$ have no common factors.
• Always multiply BOTH the numerator and the denominator by the same factor to keep the fraction's value unchanged.
• Check that the final answer cannot be simplified further.

Common Mistakes

• Subtracting only the coefficients (e.g., writing $\frac{-1}{3x}$) without adjusting the numerators correctly.
• Forgetting to multiply the numerator of the first fraction by 5, or the second by 2.
• Writing the denominator as $2x - 5x$ instead of using a common denominator.

Things to Be Careful About

• Ensure the final answer is a single fraction in its simplest form. Do not leave it as two separate fractions or unsimplified expressions like $\frac{1}{10x}$ written incorrectly.

Walkthrough

Here the denominators are $x$ and $x - 3$. These are different linear expressions, so the lowest common denominator is their product: $x(x - 3)$.

To change the denominator of the first fraction from $x$ to $x(x - 3)$, we must multiply by $(x - 3)$. We must do the same to the numerator: $4 \times (x - 3)$.

To change the denominator of the second fraction from $x - 3$ to $x(x - 3)$, we multiply by $x$. Again, we must multiply the numerator by $x$: $7 \times x = 7x$.

Now we add the two new numerators together: $4(x - 3) + 7x$. It is crucial to expand the bracket $4(x - 3)$ to $4x - 12$ before combining. Then we add $7x$ to $4x - 12$ to get $11x - 12$. The final fraction is this combined numerator over the common denominator.

Key Takeaways

• When denominators are distinct linear factors (like $x$ and $x-3$), the common denominator is their product.
• Always expand brackets in the numerator fully before collecting like terms.
• The final answer is usually accepted in factored denominator form $x(x-3)$ rather than expanded form $x^2 - 3x$.

Common Mistakes

• Forgetting to distribute the multiplier to the numerator of the first term (e.g., writing $\frac{4 + 7x}{x(x - 3)}$).
• Failing to expand $4(x - 3)$ correctly (e.g., writing $4x - 3$).
• Collecting terms incorrectly (e.g., adding constants where there are none, or mixing up $x$ terms).

Things to Be Careful About

• Show the expansion step clearly ($4(x-3) = 4x-12$) to secure method marks.
• The question asks for a single fraction; do not split it back into partial fractions.

Walkthrough

Function notation $f(x)$ tells us what operation to perform on the input $x$. Here, we double the input, subtract 3, and then divide by 4. When asked for $f(2)$, we replace every instance of $x$ with the number 2. So $2x$ becomes $2 \times 2 = 4$. Then $4 - 3 = 1$. Finally, we divide by 4 to get $\frac{1}{4}$.

Key Takeaways

• Substitution means replacing the variable with the given value.
• Use parentheses when substituting negative numbers or complex expressions, though here $x=2$ is positive and simple.
• Answers can often be given as fractions or decimals unless specified otherwise.

Common Mistakes

• Multiplying incorrectly (e.g., $2 \times 2 = 5$).
• Subtracting in the wrong order (e.g., $3 - 4$).
• Dividing only part of the numerator (e.g., thinking the 4 divides only the 3). Remember the fraction bar acts as a grouping symbol for the entire numerator.

Things to Be Careful About

• Ensure you calculate the entire numerator ($2x - 3$) before dividing by the denominator (4). A common error is to write $2(2) - \frac{3}{4}$ instead of $\frac{2(2)-3}{4}$.

Walkthrough

The inverse function essentially reverses the operations of the original function. The original function multiplies by 2, subtracts 3, and divides by 4. The inverse should do the opposite in reverse order: multiply by 4, add 3, and divide by 2.

Mathematically, we set $y = \frac{2x - 3}{4}$. To find the inverse, we treat $x$ as the output and solve for the input $y$. Swapping variables gives $x = \frac{2y - 3}{4}$. We then perform algebraic steps to isolate $y$: multiply by 4 ($4x = 2y - 3$), add 3 ($4x + 3 = 2y$), and divide by 2 ($y = \frac{4x + 3}{2}$). Finally, we simplify the fraction to get the linear form $2x + 1.5$. By comparing this to $cx + d$, we identify $c=2$ and $d=1.5$.

Key Takeaways

• To find an inverse function algebraically: replace $f(x)$ with $y$, swap $x$ and $y$, then solve for $y$.
• The inverse of a linear function $y = mx + c$ is also linear.
• Be comfortable manipulating fractions and rearranging equations.

Common Mistakes

• Forgetting to swap $x$ and $y$ before solving.
• Algebra errors when rearranging, such as subtracting 3 instead of adding it, or forgetting to divide the constant term (+3) by 2.
• Leaving the answer as a single fraction $\frac{4x+3}{2}$ when the question asks for the form $cx+d$, leading to difficulty identifying $c$ and $d$.

Things to Be Careful About

• The question asks for $c$ and $d$ specifically. Make sure to explicitly state their values.
• $d$ is $1.5$ (or $\frac{3}{2}$), not $3$ or $1$. Don't forget to divide the constant term by the coefficient of $y$.

Walkthrough

The problem states that when we put $g$ into the function, the output is $-g$ (the negative of the input). So we write the expression for $f(g)$ using the rule $\frac{2x-3}{4}$, which becomes $\frac{2g-3}{4}$. We equate this to $-g$. Now we have an equation to solve for $g$. First, remove the fraction by multiplying everything by 4. This gives $2g - 3 = -4g$. Next, move all terms with $g$ to one side by adding $4g$ to both sides, giving $6g - 3 = 0$. Then move the constant to the other side ($6g = 3$) and divide by the coefficient of $g$ ($6$) to find $g = \frac{3}{6} = \frac{1}{2}$.

Key Takeaways

• Function notation $f(g)$ just means 'replace x with g'.
• Equations involving fractions can be simplified by multiplying through by the denominator.
• When collecting variables, pay attention to signs (adding $4g$ cancels out $-4g$).

Common Mistakes

• Multiplying only part of the left-hand side by 4 (forgetting that the equals sign applies to both sides).
• Sign errors when moving $-4g$ to the left (writing $2g - 4g$ instead of $2g + 4g$).
• Arithmetic errors in simplifying $\frac{3}{6}$.

Things to Be Careful About

• Ensure you show the equation $\frac{2g - 3}{4} = -g$ clearly, as this earns a method mark.
• Double-check your solution by substituting $g = 0.5$ back into the original function: $f(0.5) = \frac{2(0.5)-3}{4} = \frac{1-3}{4} = \frac{-2}{4} = -0.5$. This matches $-g$, confirming the answer.