Standard +0.3 This question involves straightforward composition of functions and solving for a constant, followed by a routine inverse operation. While it requires multiple steps (finding a, then solving gf(x)=68), each step uses standard techniques: substitution into composite functions, basic algebra, and cubing. The conceptual demand is low—no novel insight needed, just careful application of function notation and inverse operations.
7 The function f is defined for all real values of \(x\) by \(\mathrm { f } ( x ) = 2 x + 5\). The function g is defined for all real values of \(x\) and is such that \(\mathrm { g } ^ { - 1 } ( x ) = \sqrt [ 3 ] { x - a }\), where \(a\) is a constant. It is given that \(\mathrm { fg } ^ { - 1 } ( 12 ) = 9\). Find the value of \(a\) and hence solve the equation \(\operatorname { gf } ( x ) = 68\).
dep *M; earned at stage \(2x + 5 = ...\); if expanding to produce cubic equation, earned with attempt at linear and quadratic factors
Obtain \(-\frac{1}{2}\)
A1
And no others; dependent on correct work throughout
OR: State or imply \(f(x) = g^{-1}(68)\)
B2
Attempt solution of equation of form \(2x + 5 = \sqrt[3]{68-4}\)
M1
Obtain \(-\frac{1}{2}\)
A1
# Question 7:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show composition of functions | M1 | The right way round; or equiv |
| Obtain $2\sqrt[3]{12-a} + 5 = 9$ | A1 | Or equiv |
| Obtain $a = 4$ | A1 | |
| **EITHER:** Attempt to find $g(x)$ | *M1 | Obtaining $px^3 + q$ or $py^3 + q$ form |
| Obtain $(2x+5)^3 + 4 = 68$ | A1ft | Following their value of $a$ |
| Attempt solution of equation | M1 | dep *M; earned at stage $2x + 5 = ...$; if expanding to produce cubic equation, earned with attempt at linear and quadratic factors |
| Obtain $-\frac{1}{2}$ | A1 | And no others; dependent on correct work throughout |
| **OR:** State or imply $f(x) = g^{-1}(68)$ | B2 | |
| Attempt solution of equation of form $2x + 5 = \sqrt[3]{68-4}$ | M1 | |
| Obtain $-\frac{1}{2}$ | A1 | |
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7 The function f is defined for all real values of $x$ by $\mathrm { f } ( x ) = 2 x + 5$. The function g is defined for all real values of $x$ and is such that $\mathrm { g } ^ { - 1 } ( x ) = \sqrt [ 3 ] { x - a }$, where $a$ is a constant. It is given that $\mathrm { fg } ^ { - 1 } ( 12 ) = 9$. Find the value of $a$ and hence solve the equation $\operatorname { gf } ( x ) = 68$.
\hfill \mbox{\textit{OCR C3 2012 Q7 [7]}}