Inverse Functions - Questions
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 \(0<x \leq 3\)
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 \(f(x)=\dfrac{1}{x+2}\)
(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 \(f(x)=-\sqrt{1-x}\)
(i) State the domain and range of \(f(x)\).
(ii) Find the inverse function \(f^{-1}(x)\) and state the domain and range.
i) Domain : \(x \leq 1\), Range : \(f(x) \leq 0\)
ii) \( f^{-1}(x)=1-x^2\), Domain : \(x \leq 0\), Range: \(f^{-1}(x) \leq 1\)
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.
(i) Domain : all \(x\), Range : all \(f(x)\)
ii) \(f^{-1}(x)=\sqrt[3]{x+1}\), Domain : all \(x\), Range : all \(f^{-1}(x)\)
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 \(-5 \leq x \leq 4\).
i) Refer to worked solution
ii) \(f^{-1}(x)=2-\sqrt{4-x} \quad\left(f^{-1}(x) \leq 2\right)\)
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 \(f(x)=(x-1)^2+2\) for \(x \geq 1\).
(i) State the range of \(f(x)\).
(ii) Find the inverse function \(f^{-1}(x)\) and state the domain of \(f^{-1}(x)\).
i) The range is \(f(x) \geq 2\)
ii) \(f^{-1}(x)=1+\sqrt{x-2}, f^{-1}(x) \geq 1\), Domain is \(x \geq 2\)
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 \(-5 \leq x \leq 4\).
i) Refer to worked solution
ii) \(f^{-1}(x)=1+\sqrt{4-x} \quad\left(f^{-1}(x) \geq 1\right)\)
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 \(f(x) = \sqrt {x + 1} \),
i) State the domain and range of \(f(x)\).
ii) Find the inverse function \({f^{ - 1}}(x)\) and state the domain \({f^{ - 1}}(x)\)
i) Domain is \(x \ge - 1\), Range is \(f(x) \ge 0\)
ii) \({f^{ - 1}}(x) = {x^2} - 1\), Domain is \(x \ge 0\)
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 \({f^{ - 1}}(x)\).
i) Domain is all \(x\), Range is \(f(x) \ge - 1\)
ii) \({f^{ - 1}}(x) = \sqrt {x + 1} \,\,\,({f^{ - 1}}(x) \ge 0)\), Domain is \(x \ge - 1\)
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 \(f^{-1}(x)\).
i) \(f'(x) > 0 \) for all \(x\) ii) \(y=-1\) iii) Refer to worked solution iv) \(f(x)\) is a monotonic increasing function v) all \(x \neq -1\)
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 \(x\).
i) Refer to worked solutions ii) Refer to worked solutions iii) \(\ln\left( \dfrac{x + \sqrt{x^2 + 4}}{2} \right)\)
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) \(x = 2\) is the vertical asymptote, \(y = f(x) = 1\) is the horizontal asymptote ii) \(2+ \dfrac{1}{x-1}\) iii) \(\text{The domain of } f^{-1}(x) \text{ is } x > 1, \text{The range of } f^{-1}(x) \text{ is } y > 2\)
