DP Mathematics: Analysis and Approaches Questionbank

Find $\left(f\circ g\right)\left(x\right)$.

Solve the equation $\left(f\circ g\right)\left(x\right)=x$.

Hence or otherwise, given that $g\left(2a\right)=f^{-1}\left(2a\right)$, find the value of $a$.

The functions $f$ and $g$ are defined for $x\in \mathrm{ℝ}$ by $f\left(x\right)=x-2$ and $g\left(x\right)=ax+b$, where $a, b\in \mathrm{ℝ}$.

Given that $\left(f\circ g\right)\left(2\right)=-3$ and $\left(g\circ f\right)\left(1\right)=5$, find the value of $a$ and the value of $b$.

State the range of $f$.

Find an expression for $f^{-1}\left(x\right)$, stating its domain.

Find the coordinates of the point(s) where the graphs of $y=f\left(x\right)$ and $y=f^{-1}\left(x\right)$ intersect.

Arc $\text{APB}$ is twice the length of chord $\left[\text{AB}\right]$.

Find the value of $\theta$.

Write down the value of $f\left(2\right)$.

Write down the value of $\left(f\circ f\right)\left(2\right)$.

Find an expression for $g^{-1}\left(x\right)$, justifying your answer.

Given that $\left(g\circ f\right)\left(x\right)\le 0$ for all $x\in \mathrm{ℝ}$, determine the set of possible values for $c$.

Write down the value of $f\prime \left(4\right)$.

Show that $a=8$.

Find $f\left(4\right)$.

Write down an expression for $f^{-1}\left(x\right)$.

Find the value of $f^{-1}\left(\sqrt{32}\right)$.

Find the value of $p$ and the value of $q$.

Show that $27, p, q$ and $125$ are four consecutive terms in a geometric sequence.

Find $h\left(4\right)$.

Hence find the equation of the tangent to the graph of $h$ at $x=4$.

The function $g$ is defined by $g\left(x\right)=\frac{ax+4}{3-x}$, where $x\in \mathrm{ℝ}, x\ne 3$ and $a\in \mathrm{ℝ}$.

Given that $g\left(x\right)=g^{-1}\left(x\right)$, determine the value of $a$.

the vertical asymptote of the graph of $f$.

the $y$-axis.

the horizontal asymptote of the graph of $f$.

the $x$-axis.

Sketch the graph of $f$ on the axes below.

Find $(f\circ g)\left(x\right)$.

Find $(f\circ g)\left(0\right)$.

Find the value of $x$ such that $f\left(x\right)=0$.

Given that $\left(h\circ g\right)\left(a\right)=\frac{\mathrm{\pi }}{4}$, find the value of $a$.

Give your answer in the form $p+\frac{q}{2}\sqrt{r}$, where $p, q, r\in {\mathrm{\mathbb{Z}}}^{+}$.

Find $f^{-1}\left(x\right)$.

Using an algebraic approach, show that the graph of $f^{-1}$ is obtained by a reflection of the graph of $f$ in the $y$-axis followed by a reflection in the $x$-axis.

Show that $\left(g\circ f\right)\left(x\right)=2x+11$.

Given that ${\left(g\circ f\right)}^{-1}\left(a\right)=4$, find the value of $a$.

Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y=f\left(x\right)$.

sketch the graph of $y=f\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

sketch the graph of $y=f^{-1}\left(x\right)$, showing clearly any axis intercepts and giving the equations of any asymptotes.

Hence, or otherwise, solve the inequality $f\left(x\right)>f^{-1}\left(x\right)$.

Find $\left(f\circ g\right)\left(\left(x\text{,}y\right)\right)$.

Find $\left(g\circ f\right)\left(\left(x\text{,}y\right)\right)$.

State with a reason whether or not $f$ and $g$ commute.

Find the inverse of $f$.

(i)     Sketch the graph of $y=f(x)$ and hence justify whether or not $f$ is a bijection.

(ii)     Show that $A$ is a group under the binary operation of multiplication.

(iii)     Give a reason why $B$ is not a group under the binary operation of multiplication.

(iv)     Find an example to show that $f(a\times b)=f(a)\times f(b)$ is not satisfied for all $a,b\in A$.

(i)     Sketch the graph of $y=g(x)$ and hence justify whether or not $g$ is a bijection.

(ii)     Show that $g(a+b)=g(a)\times g(b)$ for all $a,b\in \mathbb{R}$.

(iii)     Given that $\{\mathbb{R},+\}$ and $\{D,\times \}$ are both groups, explain whether or not they are isomorphic.

Find $(g\circ f)(x)$.

Given that $\underset{x\to +\mathrm{\infty }}{lim}(g\circ f)(x)=-3$, find the value of $b$.

Show that $(f\circ g)(x)=x^4-4x^2+3$.

Find $h\left(x\right)$.

Let $\text{C}$ be a point on the graph of $h$. The tangent to the graph of $h$ at $\text{C}$ is parallel to the graph of $f$.

Find the $x$-coordinate of $\text{C}$.