IB Mathematics AI HL 2.1 Functions Question Bank

Practise IB Mathematics AI HL 2.1 by analysing advanced function models, transformations, inverse and regression evidence with technology.

Exam points

Analyse parameterised, composite, piecewise and inverse functions with domain and range restrictions.
Use technology to fit, compare and validate exponential, logarithmic, polynomial and contextual models.
Interpret intersections, residuals, asymptotes and sensitivity while checking model assumptions.

Covered syllabus points

SL 2.1—Straight lines SL 2.2—Functions and inverses SL 2.3—Graphing functions SL 2.4—Key graph features SL 2.5—Modelling with functions SL 2.6—Modelling process

[Maximum number: 5]

Line $L_{1}$ is tangent to the graph of a function f(x) at the point P(3,-1). Line $L_{2}$ is given by the equation $y=-\frac{1}{2} x-\frac{5}{2}$ and is perpendicular to $L_{1}$ .

(a)

Write down the gradient of $L_{1}$ .

[ 1 ]

(b)

Find the equation of $L_{1}$ in the form y=m x+c.

[ 2 ]

attempt to substitute their part (a) and point (3,-1) into the slope-intercept form or point-slope form of an equation
$-1=2 \times 3+c \quad$ OR $\quad y+1=2(x-3)$
y=2 x-7
Note: Equation must be in the form y=m x+c for A1 to be awarded.

(c)

Show that $L_{2}$ is not the line that is normal to f(x) at point P .

[ 2 ]

METHOD 1
attempt to show that P does not lie on $L_{2}$
e.g. $-\frac{1}{2}(3)-\frac{5}{2} \quad \mathbf{OR}$ graph showing $L_{2}$ and P in approximate correct locations
$-1 \neq-\frac{1}{2}(3)-\frac{5}{2}(-1 \neq-4) \mathbf{OR}(3,-1)$ does not lie on the graph of $L_{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 2
attempt to find the equation of the normal line at (3,-1)
$\left(-1=-\frac{1}{2}(3)+c \quad\right.$ OR $\left.\quad y+1=-\frac{1}{2}(x-3)\right)$
the normal line is $y=-\frac{1}{2} x+\frac{1}{2}$
hence $L_{2}$ is not the normal line to f(x) at point P

METHOD 3
attempt to find the intersection of $L_{1}$ and $L_{2}$

Intersection of y=2 x-7 and $y=-\frac{1}{2} x-\frac{5}{2}$ is (1.8,-3.4)
$x=1.8 \neq 3 \quad$ OR $\quad y=-3.4 \neq-1$
hence $L_{2}$ is not the normal line to f(x) at point P

Note: Accept equivalent written arguments provided values are seen.
Methods 1 and 2 are independent of the answers in (a) and (b) but FT marks can be given for Method 3.

[Maximum number: 2]

The function f is defined by $f(x)=4 \ln (2 x-3)$ , where $x>\frac{3}{2}$ .
The graph of y=f(x) is obtained from the graph of $y=\ln x$ by a sequence of three transformations.

The graph of y=g(x) is shown.
The graph passes through the points (-2,-2),(-1,0),(0,2),(1,3),(2,5) and (3,6).

Figure for Question 2.1 Functions - SL content question 2 — IB Maths AI HL

Find $(f \circ g)(2)$ .

$\quad g(2)=5$ $f(5)=7.78 \quad(=7.78364 \ldots, 4 \ln 7)$

[Maximum number: 3]

A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.

k=A \mathrm{e}^{-\frac{c}{T}}

This equation links a variable k with the temperature T, where A and c are positive constants and T>0.

Given that $\lim_{T\to\infty}k=A$ and $\lim_{T\to0}k=0$ , sketch the graph of k against T.

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Covered syllabus points

2.1 Functions - SL content question 1

Question (a)

Question (b)

Question (c)

2.1 Functions - SL content question 2

2.1 Functions - SL content question 3