Consider the function \(f(x) = \sqrt {x + 1} \),

i) State the domain and range of \(f(x)\).

ii) Find the inverse function \({f^{ - 1}}(x)\) and state the domain \({f^{ - 1}}(x)\)

Consider the function \(f(x) = {x^2} - 1\),

i) State domain and range \(f(x)\).

ii) For \(f(x) = {x^2} - 1\) and \(x \ge 0\), find \({f^{ - 1}}(x)\) and state the domain of \({f^{ - 1}}(x)\).

Consider the function \(f(x) = 1+\dfrac{1}{x-2}\) for \(x \geq 2\).

i) Give the equations of the horizontal and vertical asymptotes for \(y = f(x)\).
ii)Find the inverse function \(f^{-1}(x)\).
iii) State the domain and range of the inverse function.

i) Show that the curve \(y = x^3 - 3x\) has stationary points at \((1, -2)\) and \((-1, 2)\).
ii) Find the largest domain including zero such that the function \(f(x) = x^3 - 3x\) has an inverse \(f^{-1}(x)\).
iii) On the same set of axes sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\).
iv) Find the gradient of the tangent to the curve \(y = f^{-1}(x)\) at the point \(\left(-\dfrac{11}{8},\, \dfrac{1}{2}\right)\).

Consider the function \(f(x) = \dfrac{x}{1-x}\).

i) Show that \(f'(x) > 0\) for all \(x\) in the domain.
ii) State the equation of the horizontal asymptote of \(y = f(x)\).
iii) Without using any further calculus, sketch the graph of \(y = f(x)\).
iv) Explain why \(f(x)\) has an inverse function \(f^{-1}(x)\).
v) Write down the domain of \(f^{-1}(x)\).

Let \(g(x) = e^{x} - \dfrac{1}{e^x}\) for all real values of \(x\).

i) Sketch the graph of \(y = g(x)\) and explain why \(g(x)\) has an inverse function for all values of \(x\).
ii) On a separate diagram, sketch the graph of the inverse function.
iii) Find an expression for \(y = f^{-1}(x)\) in terms of \(x\).

Consider the function \(f(x)=2^x\),

(i) Sketch the function in the domain \(-2 \leq x \leq 2\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(0<x \leq 4\)

Consider the function \(f(x)=x(x-2)\) for \(x \geq 1\).

(i) Sketch the function in the domain \(1 \leq x \leq 4\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(-1<x \leq 4\).

Consider the function \(f(x)=\dfrac{1}{x+2}\)

(i) Sketch the function in the domain \(-4 \leq x \leq 2\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(-4<x \leq 2\).

Consider the function \(f(x)=-\dfrac{2}{3} x+1\),

(i) Find the inverse function \(f^{-1}(x)\).

(ii) Let \(f^{-1}(x)=g(x)\), show that \(f(g(x))=g(f(x))=x\)

Consider the function \(f(x)=-\sqrt{1-x}\)

(i) State the domain and range of \(f(x)\).

(ii) Find the inverse function \(f^{-1}(x)\) and state the domain and range.

Consider the function \(f(x)=x^3-1\)

(i) State the domain and range of \(f(x)\).

(ii) Find the inverse function \(f^{-1}(x)\) and state the domain and range.

Consider the function \(f(x)=\log _2(x-1)\)

(i) Sketch the function in the domain \(1<x \leq 5\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch the function on the same graph in the domain \(-5 \leq x \leq 2\).

Consider the function \(f(x)=x(4-x)\)

(i) Sketch the function in the domain \(-1<x \leq 2\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch the function on the same graph in the domain \(-5 \leq x \leq 4\).

Consider the function \(f(x)=\dfrac{x-1}{x+1}\)

(i) Sketch the function in the domain \(-4<x \leq 4\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch the function on the same graph in the domain \(-4 \leq x \leq 4\)

Consider the function \(f(x)=\log _2(x+1)\)

(i) Find the inverse function \(f^{-1}(x)\).

(ii) Let \(f^{-1}(x)=g(x)\), show that \(f(g(x))=g(f(x))=x\)

Consider the function \(f(x)=\dfrac{1}{x-2}\),

(i) Find the inverse function \(f^{-1}(x)\).

(ii) Find the value of \(x\) for which \(f(x)=f^{-1}(x)\)

Consider the function \(f(x)=(x-1)^2+2\) for \(x \geq 1\).

(i) State the range of \(f(x)\).

(ii) Find the inverse function \(f^{-1}(x)\) and state the domain of \(f^{-1}(x)\).

Consider the function \(f(x)=\dfrac{2+3 e^x}{3}\),

(i) Sketch the function in the domain \(-4 \leq x \leq 4\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(0<x \leq 4\).

Consider the function \(f(x)=\dfrac{x}{x+1}\)

(i) Sketch the function in the domain \(-4<x \leq 4\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch the function on the same graph in the domain \(-4 \leq x \leq 4\)

Consider the function \(f(x)=(x+1)(3-x), x \geq 1\)

(i) Sketch the function in the domain \(1<x \leq 4\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch the function on the same graph in the domain \(-5 \leq x \leq 4\).

Consider the function \(f(x)=\dfrac{1}{x-2}\),

(i) Find the inverse function \(f^{-1}(x)\).

(ii) Let \(f^{-1}(x)=g(x)\), show that \(f(g(x))=g(f(x))=x\)

Consider the function \(f(x)=\sqrt{4-x}\).

(i) State the range of \(f(x)\).

(ii) Find the inverse function \(f^{-1}(x)\) and state the domain of \(f^{-1}(x)\).

Consider the function \(f(x)=2-x^2, x \geq 0\).

(i) Find the inverse function \(f^{-1}(x)\).

(ii) Find the value of \(x\) for which \(f(x)=f^{-1}(x)\)

Consider the function \(f(x)=\log _2(x+2)\),

(i) Sketch the function in the domain \(-3 \leq x \leq 3\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(-3<x \leq 3\)

Consider the function \(f(x)=(x-1)^2, x \leq 1\)

(i) Sketch the function in the domain \(-1 \leq x \leq 3\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(-1<x \leq 3\)

Consider the function \(f(x)=\dfrac{1}{(x-1)^2}, x<1\)

(i) Sketch the function in the domain \(-3 \leq x \leq 1\).

(ii) Find the inverse function \(f^{-1}(x)\) and sketch this function on the same graph in the domain \(0<x \leq 3\)

The functions \(f(x) = 4 - {x^2}\) and \(g(x) = \dfrac{1}{{4 - x}}\) are

The inverse function of \(f(x) = 2x - 1\) is 

\(f(x) = \sqrt {x - 1} \) ,the domain of its inverse is?

\(f(x) = {\log _e}(x - 2)\) , the range of its inverse is?

\({\rm{For }}f(x) = {x^2} - 2x + 2{\rm{ , }}\,x \ge 1\) the inverse is?