DP Mathematics SL Questionbank

• Syllabus

2.1

Sub sections and their related questions

Concept of function \(f:x \mapsto f\left( x \right)\) .

• 08N.1.sl.TZ0.4b: Write down the range of \({f^{ - 1}}\) .
• 08M.1.sl.TZ2.5b(i) and (ii): Let \(g(x) = f(x + 3)\) . (i)     Find \(g( - 3)\) . (ii)    Describe fully the transformation...
• 09N.1.sl.TZ0.1a: (i)     Find \(g(0)\) . (ii)    Find \((f \circ g)(0)\) .
• SPNone.2.sl.TZ0.10a(i), (ii) and (iii): Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable...
• 13M.1.sl.TZ1.10a: Find the value of \(f(0)\) .
• 13M.2.sl.TZ1.9a: Write down \(f(0)\) .
• 13M.2.sl.TZ1.9c: Find the range of \(f\) .
• 13M.1.sl.TZ2.4a.i: Write down the value of \(f(2)\).
• 13M.1.sl.TZ2.10a: Write down the value of \(g(3)\) , of \(f'(3)\) , and of \(h''(2)\) .
• 13M.1.sl.TZ2.10c: find the \(y\)-coordinate of P.
• 14M.1.sl.TZ2.3a(ii): Write down the value of  \({f^{ - 1}}(1)\).
• 14M.1.sl.TZ2.3a(i): Write down the value of \(f( - 3)\).
• 14M.1.sl.TZ2.3b: Find the domain of \({f^{ - 1}}\).
• 16M.1.sl.TZ1.1a: Write down \(g(2)\).
• 16M.1.sl.TZ1.1b: Find \((f \circ g)(x)\).
• 16M.1.sl.TZ1.1c: Find \({f^{ - 1}}(x)\).
• 17M.2.sl.TZ2.3a: Write down the range of \(f\).
• 17M.2.sl.TZ2.3b: On the grid above, sketch the graph of \(g\).
• 17M.2.sl.TZ2.3c: Write down the domain of \(g\).
• 17N.1.sl.TZ0.3a: Write down the range of \(f\).
• 17N.1.sl.TZ0.3b.i: Write down \(f(2)\);
• 17N.1.sl.TZ0.3b.ii: Write down \({f^{ - 1}}(2)\).
• 17N.1.sl.TZ0.3c: On the grid, sketch the graph of \({f^{ - 1}}\).
• 18M.1.sl.TZ1.1a: Write down f (14).
• 18M.1.sl.TZ1.1b: Find \(\left( {g \circ f} \right)\) (14).
• 18M.1.sl.TZ1.1c: Find g−1(x).
• 18M.1.sl.TZ1.3a.i: Write down the value of f (0).
• 18M.1.sl.TZ1.3a.ii: Write down the value of f −1 (1).
• 18M.1.sl.TZ1.3b: Write down the range of f −1.
• 18M.1.sl.TZ1.3c: On the grid above, sketch the graph of f −1.

Domain, range; image (value).

• 08N.1.sl.TZ0.4b: Write down the range of \({f^{ - 1}}\) .
• 10M.1.sl.TZ1.7a: Show that \({f^{ - 1}}(x) = {3^{2x}}\) .
• 10M.1.sl.TZ1.7b: Write down the range of \({f^{ - 1}}\) .
• 10M.1.sl.TZ1.7c: Let \(g(x) = {\log _3}x\) , for \(x > 0\) . Find the value of \(({f^{ - 1}} \circ g)(2)\) ,...
• 09M.1.sl.TZ1.6a: (i)     Show that \({f^{ - 1}}(x) = \ln x - 3\) . (ii)    Write down the domain of \({f^{ - 1}}\) .
• 09M.2.sl.TZ1.10e: Write down the range of values for the gradient of \(f\) .
• SPNone.2.sl.TZ0.10a(i), (ii) and (iii): Jane thinks that the function \(f(t) = - 0.25{t^3} - 2.32{t^2} + 1.93t + 106\) is a suitable...
• 13M.1.sl.TZ1.5a: Find \({f^{ - 1}}(2)\) .
• 13M.1.sl.TZ1.10a: Find the value of \(f(0)\) .
• 13M.2.sl.TZ1.9a: Write down \(f(0)\) .
• 13M.2.sl.TZ1.9c: Find the range of \(f\) .
• 13M.1.sl.TZ2.4a.i: Write down the value of \(f(2)\).
• 13M.1.sl.TZ2.10a: Write down the value of \(g(3)\) , of \(f'(3)\) , and of \(h''(2)\) .
• 13M.1.sl.TZ2.10c: find the \(y\)-coordinate of P.
• 13M.1.sl.TZ2.4a.ii: Write down the value of \({f^{ - 1}}( - 1)\) .
• 14M.1.sl.TZ2.3a(ii): Write down the value of  \({f^{ - 1}}(1)\).
• 14M.1.sl.TZ2.3a(i): Write down the value of \(f( - 3)\).
• 14M.1.sl.TZ2.3b: Find the domain of \({f^{ - 1}}\).
• 13N.1.sl.TZ0.8e: Given that \({h^{ - 1}}(a) = 3\), find the value of \(a\).
• 16M.1.sl.TZ1.1a: Write down \(g(2)\).
• 16M.1.sl.TZ1.1b: Find \((f \circ g)(x)\).
• 16M.1.sl.TZ1.1c: Find \({f^{ - 1}}(x)\).
• 16N.2.sl.TZ0.1a: Find \(f(8)\).
• 17M.2.sl.TZ2.3a: Write down the range of \(f\).
• 17M.2.sl.TZ2.3b: On the grid above, sketch the graph of \(g\).
• 17M.2.sl.TZ2.3c: Write down the domain of \(g\).
• 17N.1.sl.TZ0.3a: Write down the range of \(f\).
• 17N.1.sl.TZ0.3b.i: Write down \(f(2)\);
• 17N.1.sl.TZ0.3b.ii: Write down \({f^{ - 1}}(2)\).
• 17N.1.sl.TZ0.3c: On the grid, sketch the graph of \({f^{ - 1}}\).
• 18M.1.sl.TZ1.1a: Write down f (14).
• 18M.1.sl.TZ1.1b: Find \(\left( {g \circ f} \right)\) (14).
• 18M.1.sl.TZ1.1c: Find g−1(x).
• 18M.1.sl.TZ1.3a.i: Write down the value of f (0).
• 18M.1.sl.TZ1.3a.ii: Write down the value of f −1 (1).
• 18M.1.sl.TZ1.3b: Write down the range of f −1.
• 18M.1.sl.TZ1.3c: On the grid above, sketch the graph of f −1.

Composite functions.

• 12M.1.sl.TZ2.2a: Find \({f^{ - 1}}(x)\) . 
• 12M.1.sl.TZ2.2b: Find \((f \circ g)(1)\) . 
• 08M.1.sl.TZ1.7b: Let \(g(x) = {{\rm{e}}^x}\) . Find \((g \circ f)(x)\) , giving your answer in the form...
• 10N.1.sl.TZ0.9a: Find \((f \circ g)(x)\) .
• 10N.1.sl.TZ0.9b: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9c: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10N.1.sl.TZ0.9d: The vector \(\left( {\begin{array}{*{20}{c}}3\\{ - 1}\end{array}} \right)\) translates the graph...
• 10M.1.sl.TZ1.7a: Show that \({f^{ - 1}}(x) = {3^{2x}}\) .
• 10M.1.sl.TZ1.7b: Write down the range of \({f^{ - 1}}\) .
• 10M.1.sl.TZ1.7c: Let \(g(x) = {\log _3}x\) , for \(x > 0\) . Find the value of \(({f^{ - 1}} \circ g)(2)\) ,...
• 10M.1.sl.TZ2.4a: Find \(f\left( {\frac{\pi }{2}} \right)\) .
• 10M.1.sl.TZ2.4b: Find \((g \circ f)\left( {\frac{\pi }{2}} \right)\) .
• 10M.1.sl.TZ2.4c: Given that \((g \circ f)(x)\) can be written as \(\cos (kx)\) , find the value of k,...
• 09N.1.sl.TZ0.1a: (i)     Find \(g(0)\) . (ii)    Find \((f \circ g)(0)\) .
• 09M.2.sl.TZ1.3a: Find an expression for \(h(x)\) .
• 09M.1.sl.TZ2.1b: Find \((f \circ g)(4)\) .
• 11N.2.sl.TZ0.1a: Find \({f^{ - 1}}(x)\) .
• 11N.2.sl.TZ0.1c: Find \((f \circ g)(3.5)\) .
• 11M.1.sl.TZ1.1a: Find \((g \circ f)(x)\) .
• 11M.1.sl.TZ1.1b: Write down \({g^{ - 1}}(x)\) .
• 11M.1.sl.TZ1.1c: Find \((f \circ {g^{ - 1}})(5)\) .
• 11M.2.sl.TZ2.1a: Find \(h(x)\) .
• 11M.2.sl.TZ2.1b: Find \({h^{ - 1}}(x)\) .
• 13M.1.sl.TZ1.5b: Let \(g\) be a function such that \({g^{ - 1}}\) exists for all real numbers. Given that...
• 13M.1.sl.TZ2.1b: Find \((f \circ g)(1)\) .
• 11N.2.sl.TZ0.1b: Find \((f \circ g)(x)\) .
• 13N.1.sl.TZ0.8b: Show that \(\left( {g \circ {f^{ - 1}}} \right)(x) = \frac{5}{{x + 2}}\).
• 14N.2.sl.TZ0.1a: Find \((f \circ g)(x)\).
• 15M.1.sl.TZ1.4b: Find \((f \circ f)( - 1)\).
• 15M.2.sl.TZ1.10c: Let \(g(x) = \ln \left( {f(x)} \right)\) and \(f(2) = 3\). Find \(g'(2)\).
• 15N.1.sl.TZ0.5b: Let \(g\) be a function so that \((f \circ g)(x) = 8{x^6}\). Find \(g(x)\).
• 16M.1.sl.TZ1.1a: Write down \(g(2)\).
• 16M.1.sl.TZ1.1b: Find \((f \circ g)(x)\).
• 16M.1.sl.TZ1.1c: Find \({f^{ - 1}}(x)\).
• 16M.1.sl.TZ2.6a: Write \(h(x)\) in the form \(a\sin (bx)\), where \(a,{\text{ }}b \in \mathbb{Z}\).
• 16M.1.sl.TZ2.6b: Hence find the range of \(h\).
• 16N.2.sl.TZ0.1b: Find \((g \circ f)(x)\).
• 16N.2.sl.TZ0.1c: Solve \((g \circ f)(x) = 0\).
• 17M.1.sl.TZ1.2a: Find \({f^{ - 1}}(x)\).
• 17M.1.sl.TZ1.2b: Find \((f \circ g)(7)\).
• 17M.2.sl.TZ2.6a: Show that \((f \circ g)(x) = {x^4} - 4{x^2} + 3\).
• 17M.2.sl.TZ2.6b: On the following grid, sketch the graph of \((f \circ g)(x)\), for...
• 17M.2.sl.TZ2.6c: The equation \((f \circ g)(x) = k\) has exactly two solutions, for...
• 17N.1.sl.TZ0.5a: Find \((g \circ f)(x)\).
• 17N.1.sl.TZ0.5b: Given that \(\mathop {\lim }\limits_{x \to  + \infty } (g \circ f)(x) =  - 3\), find the value of...
• 18M.1.sl.TZ2.10a.i: Write down \(f'\left( 2 \right)\).
• 18M.1.sl.TZ2.10a.ii: Find \(f\left( 2 \right)\).
• 18M.1.sl.TZ2.10b: Show that the graph of g has a gradient of 6 at P.
• 18M.1.sl.TZ2.10c: Let L2 be the tangent to the graph of g at P. L1 intersects L2 at the point Q. Find the...

Identity function. Inverse function \({f^{ - 1}}\) .

• 12M.1.sl.TZ2.2a: Find \({f^{ - 1}}(x)\) . 
• 12M.1.sl.TZ2.2b: Find \((f \circ g)(1)\) . 
• 08N.1.sl.TZ0.4c: Find \({f^{ - 1}}(x)\) .
• 08M.1.sl.TZ1.7a: Find \({f^{ - 1}}(x)\) .
• 12M.1.sl.TZ1.9a: Find p .
• 12M.1.sl.TZ1.9b: The graph of f is reflected in the line \(y = x\) to give the graph of g . (i)     Write down...
• 12M.1.sl.TZ1.9c: The graph of \(f\) is reflected in the line \(y = x\) to give the graph of \(g\) . Find \(g(x)\) .
• 10M.1.sl.TZ1.7a: Show that \({f^{ - 1}}(x) = {3^{2x}}\) .
• 10M.1.sl.TZ1.7b: Write down the range of \({f^{ - 1}}\) .
• 10M.1.sl.TZ1.7c: Let \(g(x) = {\log _3}x\) , for \(x > 0\) . Find the value of \(({f^{ - 1}} \circ g)(2)\) ,...
• 09N.1.sl.TZ0.1b: Find \({f^{ - 1}}(x)\) .
• 09N.1.sl.TZ0.7a: Given that \({f^{ - 1}}(1) = 8\) , find the value of \(k\) .
• 09N.1.sl.TZ0.7b: Find \({f^{ - 1}}\left( {\frac{2}{3}} \right)\) .
• 09M.1.sl.TZ1.6a: (i)     Show that \({f^{ - 1}}(x) = \ln x - 3\) . (ii)    Write down the domain of \({f^{ - 1}}\) .
• 09M.1.sl.TZ2.1a: Find \({g^{ - 1}}(x)\) .
• SPNone.2.sl.TZ0.9a: Find \({h^{ - 1}}(x)\) .
• 11N.2.sl.TZ0.1a: Find \({f^{ - 1}}(x)\) .
• 11N.2.sl.TZ0.1c: Find \((f \circ g)(3.5)\) .
• 11M.1.sl.TZ1.1a: Find \((g \circ f)(x)\) .
• 11M.1.sl.TZ1.1b: Write down \({g^{ - 1}}(x)\) .
• 11M.1.sl.TZ1.1c: Find \((f \circ {g^{ - 1}})(5)\) .
• 11M.2.sl.TZ2.1a: Find \(h(x)\) .
• 11M.2.sl.TZ2.1b: Find \({h^{ - 1}}(x)\) .
• 13M.1.sl.TZ1.5a: Find \({f^{ - 1}}(2)\) .
• 13M.1.sl.TZ2.1a: Find \({f^{ - 1}}(x)\) .
• 11N.2.sl.TZ0.1b: Find \((f \circ g)(x)\) .
• 13M.1.sl.TZ2.4a.ii: Write down the value of \({f^{ - 1}}( - 1)\) .
• 14M.1.sl.TZ2.3a(ii): Write down the value of  \({f^{ - 1}}(1)\).
• 14M.1.sl.TZ2.3a(i): Write down the value of \(f( - 3)\).
• 14M.1.sl.TZ2.3b: Find the domain of \({f^{ - 1}}\).
• 13N.1.sl.TZ0.8a: Find \({f^{ - 1}}(x)\).
• 13N.1.sl.TZ0.8e: Given that \({h^{ - 1}}(a) = 3\), find the value of \(a\).
• 15M.1.sl.TZ1.4a: Find \({f^{ - 1}}( - 1)\).
• 15N.1.sl.TZ0.5a: Find \({f^{ - 1}}(x)\).
• 16M.1.sl.TZ1.1a: Write down \(g(2)\).
• 16M.1.sl.TZ1.1b: Find \((f \circ g)(x)\).
• 16M.1.sl.TZ1.1c: Find \({f^{ - 1}}(x)\).
• 17M.1.sl.TZ1.2a: Find \({f^{ - 1}}(x)\).
• 17M.1.sl.TZ1.2b: Find \((f \circ g)(7)\).
• 17M.2.sl.TZ1.10b.i: Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).
• 17M.2.sl.TZ1.10b.ii: Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).
• 17M.2.sl.TZ1.10c: Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\)....
• 17N.1.sl.TZ0.3a: Write down the range of \(f\).
• 17N.1.sl.TZ0.3b.i: Write down \(f(2)\);
• 17N.1.sl.TZ0.3b.ii: Write down \({f^{ - 1}}(2)\).
• 17N.1.sl.TZ0.3c: On the grid, sketch the graph of \({f^{ - 1}}\).
• 18M.1.sl.TZ1.3a.i: Write down the value of f (0).
• 18M.1.sl.TZ1.3a.ii: Write down the value of f −1 (1).
• 18M.1.sl.TZ1.3b: Write down the range of f −1.
• 18M.1.sl.TZ1.3c: On the grid above, sketch the graph of f −1.