9 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by
$$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = 2 x - 1$$
for all real values of \(x\).
- State the ranges of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\).
Explain why \(\mathrm { f } ( x )\) has no inverse.
- Find an expression for the inverse function \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
Sketch the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\) on the same axes.
- Find expressions for \(\operatorname { gf } ( x )\) and \(\operatorname { fg } ( x )\).
- Solve the equation \(\operatorname { gf } ( x ) = \mathrm { fg } ( x )\).
Sketch the graphs of \(y = \operatorname { gf } ( x )\) and \(y = \operatorname { fg } ( x )\) on the same axes to illustrate your answer.
- Show that the equation \(\mathrm { f } ( x + a ) = \mathrm { g } ^ { 2 } ( x )\) has no solution if \(a > \frac { 1 } { 4 }\).