6. Functions

Answer: $-4$

The vertical asymptote sits where the denominator is zero: $2 x + 8 = 0$, so $x = -4$. (The horizontal one is $y = 1$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $- \frac{5}{2}$
• Gave the horizontal asymptote instead → $1$
• Forgot the sign when solving → $4$

Answer: $\frac{x - 4}{x + 8}$

Factorise both: $\dfrac{\left(x - 4\right) \left(x + 3\right)}{\left(x + 3\right) \left(x + 8\right)}$. The factor $x + 3$ is common to both and cancels, leaving $\frac{x - 4}{x + 8}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{- x - 12}{11 x + 24}$
• Cancelled the wrong bracket → $\frac{x + 3}{x + 8}$

Answer: $36 x^{2} - 24 x + 8$

Square the bracket FIRST, then multiply by $4$, then add the constant — the order matters. $\left(3 x - 1\right)^2 = 9 x^{2} - 6 x + 1$, and the whole thing comes to $36 x^{2} - 24 x + 8$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $144 x^{2} - 96 x + 20$
• Squared each term inside the bracket separately → $36 x^{2} + 8$

Answer: $- \frac{4}{3}$

Multiply both sides by $x + 1$: $2 x + 1 = 5\left(x + 1\right)$. Expand, collect and divide: $x = - \frac{4}{3}$. (Check it is not $x = -1$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $2$
• Set the denominator to zero, which is the EXCLUDED value → $-1$

Answer: $-3$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-1} = -1$. So $x + 2 = -1$ and $x = -3$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $- \frac{7}{3}$
• Took the cube root but lost the sign → $-1$

Answer: $- \frac{3}{2}$

The vertical asymptote sits where the denominator is zero: $2 x + 3 = 0$, so $x = - \frac{3}{2}$. (The horizontal one is $y = -1$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $1$
• Gave the horizontal asymptote instead → $-1$
• Forgot the sign when solving → $\frac{3}{2}$

Answer: $\frac{x + 3}{x + 4}$

Factorise both: $\dfrac{\left(x + 3\right) \left(x + 7\right)}{\left(x + 4\right) \left(x + 7\right)}$. The factor $x + 7$ is common to both and cancels, leaving $\frac{x + 3}{x + 4}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{10 x + 21}{11 x + 28}$
• Cancelled the wrong bracket → $\frac{x + 7}{x + 4}$

Answer: $100 x^{2} - 40 x + 2$

Square the bracket FIRST, then multiply by $4$, then add the constant — the order matters. $\left(5 x - 1\right)^2 = 25 x^{2} - 10 x + 1$, and the whole thing comes to $100 x^{2} - 40 x + 2$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $400 x^{2} - 160 x + 14$
• Squared each term inside the bracket separately → $100 x^{2} + 2$

Answer: $-2$

Multiply both sides by $x - 3$: $5 x = 2\left(x - 3\right)$. Expand, collect and divide: $x = -2$. (Check it is not $x = 3$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $\frac{2}{5}$
• Set the denominator to zero, which is the EXCLUDED value → $3$

Answer: $-1$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-64} = -4$. So $x - 3 = -4$ and $x = -1$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $- \frac{55}{3}$
• Took the cube root but lost the sign → $7$

Answer: $-6$

The vertical asymptote sits where the denominator is zero: $x + 6 = 0$, so $x = -6$. (The horizontal one is $y = -2$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $3$
• Gave the horizontal asymptote instead → $-2$
• Forgot the sign when solving → $6$

Answer: $\frac{x - 2}{x + 4}$

Factorise both: $\dfrac{\left(x - 2\right) \left(x + 6\right)}{\left(x + 4\right) \left(x + 6\right)}$. The factor $x + 6$ is common to both and cancels, leaving $\frac{x - 2}{x + 4}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{2 x - 6}{5 x + 12}$
• Cancelled the wrong bracket → $\frac{x + 6}{x + 4}$

Answer: $100 x^{2} - 40 x$

Square the bracket FIRST, then multiply by $4$, then add the constant — the order matters. $\left(5 x - 1\right)^2 = 25 x^{2} - 10 x + 1$, and the whole thing comes to $100 x^{2} - 40 x$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $400 x^{2} - 160 x + 12$
• Squared each term inside the bracket separately → $100 x^{2}$

Answer: $\frac{8}{3}$

Multiply both sides by $x$: $6 x - 8 = 3\left(x\right)$. Expand, collect and divide: $x = \frac{8}{3}$. (Check it is not $x = 0$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $\frac{11}{6}$
• Set the denominator to zero, which is the EXCLUDED value → $0$

Answer: $-7$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-64} = -4$. So $x + 3 = -4$ and $x = -7$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $- \frac{73}{3}$
• Took the cube root but lost the sign → $1$

Answer: $- \frac{3}{2}$

The vertical asymptote sits where the denominator is zero: $2 x + 3 = 0$, so $x = - \frac{3}{2}$. (The horizontal one is $y = \frac{3}{2}$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $- \frac{7}{3}$
• Gave the horizontal asymptote instead → $\frac{3}{2}$
• Forgot the sign when solving → $\frac{3}{2}$

Answer: $\frac{x + 3}{x + 5}$

Factorise both: $\dfrac{x \left(x + 3\right)}{x \left(x + 5\right)}$. The factor $x$ is common to both and cancels, leaving $\frac{x + 3}{x + 5}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{3}{5}$
• Cancelled the wrong bracket → $\frac{x}{x + 5}$

Answer: $125 x^{2} - 200 x + 72$

Square the bracket FIRST, then multiply by $5$, then add the constant — the order matters. $\left(5 x - 4\right)^2 = 25 x^{2} - 40 x + 16$, and the whole thing comes to $125 x^{2} - 200 x + 72$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $625 x^{2} - 1000 x + 392$
• Squared each term inside the bracket separately → $125 x^{2} + 72$

Answer: $- \frac{17}{3}$

Multiply both sides by $x + 2$: $2 x - 7 = 5\left(x + 2\right)$. Expand, collect and divide: $x = - \frac{17}{3}$. (Check it is not $x = -2$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $6$
• Set the denominator to zero, which is the EXCLUDED value → $-2$

Answer: $3$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-8} = -2$. So $x - 5 = -2$ and $x = 3$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $\frac{7}{3}$
• Took the cube root but lost the sign → $7$

Answer: $-2$

The vertical asymptote sits where the denominator is zero: $x + 2 = 0$, so $x = -2$. (The horizontal one is $y = -1$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $1$
• Gave the horizontal asymptote instead → $-1$
• Forgot the sign when solving → $2$

Answer: $\frac{x + 4}{x - 3}$

Factorise both: $\dfrac{\left(x + 4\right) \left(x + 5\right)}{\left(x - 3\right) \left(x + 5\right)}$. The factor $x + 5$ is common to both and cancels, leaving $\frac{x + 4}{x - 3}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{9 x + 20}{2 x - 15}$
• Cancelled the wrong bracket → $\frac{x + 5}{x - 3}$

Answer: $20 x^{2} + 100 x + 132$

Square the bracket FIRST, then multiply by $5$, then add the constant — the order matters. $\left(2 x + 5\right)^2 = 4 x^{2} + 20 x + 25$, and the whole thing comes to $20 x^{2} + 100 x + 132$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $100 x^{2} + 500 x + 632$
• Squared each term inside the bracket separately → $20 x^{2} + 132$

Answer: $- \frac{1}{2}$

Multiply both sides by $x - 5$: $6 x - 8 = 2\left(x - 5\right)$. Expand, collect and divide: $x = - \frac{1}{2}$. (Check it is not $x = 5$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $\frac{5}{3}$
• Set the denominator to zero, which is the EXCLUDED value → $5$

Answer: $-9$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-27} = -3$. So $x + 6 = -3$ and $x = -9$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $-15$
• Took the cube root but lost the sign → $-3$

Answer: $-3$

The vertical asymptote sits where the denominator is zero: $2 x + 6 = 0$, so $x = -3$. (The horizontal one is $y = - \frac{1}{2}$, the ratio of the leading coefficients.)

Common mistakes, and the answer each one gives:

• Set the NUMERATOR to zero (that gives the x-intercept) → $8$
• Gave the horizontal asymptote instead → $- \frac{1}{2}$
• Forgot the sign when solving → $3$

Answer: $\frac{x - 2}{x - 5}$

Factorise both: $\dfrac{\left(x - 7\right) \left(x - 2\right)}{\left(x - 7\right) \left(x - 5\right)}$. The factor $x - 7$ is common to both and cancels, leaving $\frac{x - 2}{x - 5}$.

Common mistakes, and the answer each one gives:

• Cancelled the $x^2$ terms, which are terms and not factors → $\frac{9 x - 14}{12 x - 35}$
• Cancelled the wrong bracket → $\frac{x - 7}{x - 5}$

Answer: $125 x^{2} - 50 x + 4$

Square the bracket FIRST, then multiply by $5$, then add the constant — the order matters. $\left(5 x - 1\right)^2 = 25 x^{2} - 10 x + 1$, and the whole thing comes to $125 x^{2} - 50 x + 4$.

Common mistakes, and the answer each one gives:

• Multiplied the bracket by the outside number before squaring → $625 x^{2} - 250 x + 24$
• Squared each term inside the bracket separately → $125 x^{2} + 4$

Answer: $-24$

Multiply both sides by $x + 7$: $4 x - 6 = 6\left(x + 7\right)$. Expand, collect and divide: $x = -24$. (Check it is not $x = -7$, which the original forbids.)

Common mistakes, and the answer each one gives:

• Multiplied only the numerator by the right-hand side → $3$
• Set the denominator to zero, which is the EXCLUDED value → $-7$

Answer: $1$

A cube is undone by a cube root, and unlike a square root it keeps the sign: $\sqrt[3]{-64} = -4$. So $x - 5 = -4$ and $x = 1$.

Common mistakes, and the answer each one gives:

• Divided by 3 instead of taking the cube root → $- \frac{49}{3}$
• Took the cube root but lost the sign → $9$