Hint
Use the stated units and precision. Reverse the rational rule while keeping its original nonzero denominator and the inverse nonzero input.
Worked solution
(a)
The original denominator excludes -7. A nonzero numerator cannot produce zero, and every nonzero output y is produced by \(\displaystyle x=\)\(\displaystyle -7 + \frac{5}{y}\).
Answer: \(x\ne -7\), range all nonzero real outputs.
(b)
From \(\displaystyle y=\frac{5}{x + 7}\) obtain \(\displaystyle x=-7 + \frac{5}{y}\). Interchange input and output, and exchange the excluded sets.
Answer: \(\displaystyle f^{-1}(x)=-7 + \frac{5}{x}\); domain \(\displaystyle x\ne 0\), range excluding -7.
(c)
\(f(-2)=1\) because the denominator becomes 5, which is nonzero.
Answer: \(-2\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternative methods and consequential reasoning. An unexplained result does not earn a method mark.
(a) · 2 marks
- 1 mark: Use the original denominator restriction and determine precisely which outputs can be attained.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
(b) · 3 marks
- 1 mark: Solve the original relationship for its input and interchange variables.
- 1 mark: Preserve the original range as the inverse domain and vice versa.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Check the answer against the original conditions.
(c) · 2 marks
- 1 mark: Evaluate the inverse at the allowed input and verify its original-function output.
- 1 mark: Give the complete requested result, using the required units, precision and boundary inclusion.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Keep intermediate values exact until the final step.
Total: 7 marks.