A sensor's output voltage VVV, in volts, is modeled by the function V(p)=7+3lnp, p>0V(p) = 7 + 3\ln p, \, p > 0V(p)=7+3lnp,p>0, where p p\,p is the internal pressure in kilopascals. The pressure is controlled by a mechanical setting x x\,x such that p(x)=12x−63x+1\displaystyle p(x) = \frac{12x - 6}{3x + 1}p(x)=3x+112x−6 for x>0.5x > 0.5x>0.5.
Find V−1(13)V^{-1}(13)V−1(13).
Use differentiation to prove that p p\,p is an increasing function for x>0.5x > 0.5x>0.5.
Find p−1(x)p^{-1}(x)p−1(x).
Determine the range of the composite function VpVpVp.
471 exam-style questions on Edexcel A Level Maths Functions and Graphs, covering 2.1 The Modulus Function, 2.2 Functions and Mappings, 2.3 Composite Functions, 2.4 Inverse Functions, 2.5 y=|f(x)| and y=f(|x|), 2.6 Combining Transformations, and 2.7 Solving Modulus Problems. Each one has a worked solution and a mark scheme showing where the marks go.