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.