learnmath.bernatchez.net - functions

IB Functions Problem 001

2026-05-21T11:55:44+00:00

The following diagram shows the graph of the function \(f(x)\) for \(-4 \leq x \leq 2\).

Diagram \(A\)

On the set of axes of diagram \(A\), sketch the graph of \(f(-x)\).

Another function, \(g\), can be written in the form \(g(x) = a \times f(x + b)\).

The following diagram shows the graph of \(g\).

Diagram \(B\)

From diagram \(B\), write down the value of \(a\) and of \(b\) for the function \(g\).

IB Functions Problem 002

2026-05-21T11:55:43+00:00

Let \(f(x) = p\,x^2 + q\,x - 4\,p\), where \(p \ne 0\).

Find the number of roots of the equation \(f(x) = 0\).

Justify your answer.

IB Functions Problem 003

2026-05-21T11:55:42+00:00

Let \(f(x) = 2\,x - 1\) and \(g(x) = 3x^2 + 2\).

Find \(f^{-1}(x)\).
Find \((f \circ g)(1)\).

IB Functions Problem 004

2026-05-21T11:55:41+00:00

The function \(f(x)\) for \(-2 \leq x \leq 3\) is shown below.

The graph of \(f\) is transformed to obtain the graph of \(g\) shown below.

Sketch the graph of \(f(-x)\) on the set of axes below.

The function \(g\) can be written in the form \(g(x) = a\,f(x + b)\).

Find the value of \(a\) and the value of \(b\).

IB Functions Problem 005

2026-05-21T11:55:40+00:00

Consider the equation \(x^2 + (k-1)x + 1 = 0\), where \(k\) is a real number.

Find the values of \(k\) for which the equation has two equal real solutions.

IB Functions Problem 006

2026-05-21T11:55:39+00:00

The quadratic function \(f(x)\), for \(0 \leq x \leq 4\), is shown below.

The curve passes through the point \(P(0; 13)\), and its vertex is the point \(V(2; 1)\).

The function can be written in the form \(f(x) = a(x-h)^2 + k\).
1. Find the value of \(h\) and the value of \(k\).
2. Show that \(a=3\).
Calculate the area bounded by the curve of \(f\), the \(x\)-axis, and the lines \(x=2\) and \(x=4\).

IB Functions Problem 007

2026-05-21T11:55:38+00:00

Let \(f(x) =\frac{x}{-2x^2 + 5x - 2}\), for \(-2 \leq x \leq 4\), \(x \ne \frac{1}{2}\), \(x\ne2\), as shown below.

The curve has a local minimum at \(A(1;1)\) and a local maximum at \(B\).

Use the quotient rule to show that \(f^\prime(x)=\frac{2x^2 - 2}{(-2x^2+5x-2)^2}\).
Hence find the coordinates of \(B\).
Given that the line \(y=k\) does not meet the curve of \(f\), find the possible values of \(k\).

IB Functions Problem 008

2026-05-21T11:55:37+00:00

Let \(f(x) = 3\ln\,x\) and \(g(x) = \ln\,5x^3\).

Write \(g(x)\) in the form \(f(x)+\ln a\) where \(a \in \mathbb{Z}^+\).
The graph of \(g\) is a transformation of the graph of \(f\). Give a full geometric description of this transformation.

IB Functions Problem 009

2026-05-21T11:55:36+00:00

A Ferris wheel at an amusement park has a diameter of 100 metres. Figure A.

Table of heights of \(P\) in metres above the ground after \(t\) minutes. Table B.

Let \(P\) be a point on the wheel. The wheel starts with \(P\) at its lowest point, at ground level.

The wheel rotates at a constant speed, counter-clockwise. One full rotation takes \(20\) minutes.

Find the height of \(P\) above the ground after:
1. \(10\) minutes.
2. \(15\) minutes.
Let \(h(t)\) be the height of \(P\) above the ground in metres after \(t\) minutes.
1. Show that \(h(8)=90.5\).
2. Find \(h(21)\).
Sketch the graph of \(h\), for \(0 \leq t \leq 40\).
Given that \(h\) can be expressed in the form \(h(t) = a\,\cos\,bt + c\), find \(a\), \(b\) and \(c\).

IB Functions Problem 010

2026-05-21T11:55:35+00:00

Let \(f(x)=p(x-q)(x-r)\).

Part of the graph of \(f\) is shown below.

It passes through the points \((-2; 0)\), \((0; -4)\) and \((4 ; 0)\).

Find the value of \(q\) and of \(r\).
Write down the equation of the axis of symmetry.
Find the value of \(p\).

IB Functions Problem 011

2026-05-21T11:55:34+00:00

Let \(f(x) = \cos\,2x\) and \(g(x) = 2x^2 - 1\).

Find \(f\left(\frac{\pi}{2}\right)\).
Find \((g \circ f)\left(\frac{\pi}{2}\right)\).
Given that \((g \circ f)\) can be written in the form \(\cos(kx)\), find the value of \(k\), \(k \in \mathbb{Z}\).

IB Functions Problem 012

2026-05-21T11:55:33+00:00

Solve \(\log_2 x + \log_2(x - 2) = 3\), for \(x > 2\).

IB Functions Problem 013

2026-05-21T11:55:32+00:00

At the end of 1972, the population of a town was \(250\ 000\).

This population increases by \(1.3\%\) per year.

Find the population at the end of 1973.
Find the population at the end of 2002.

IB Functions Problem 014

2026-05-21T11:55:31+00:00

Let \(f(x) = \sqrt{x + 4}\), \(x \geq -4\) and \(g(x) = x^2\), \(x \in \mathbb{R}\).

Find \((g \circ f)(3)\).
Find \(f^{-1}(x)\).
Write down the domain of \(f^{-1}\).

IB Functions Problem 015

2026-05-21T11:55:30+00:00

Consider two different quadratic functions of the form \(f(x) = 4x^2 - qx + 25\).

The graph of each function has its vertex on the \(x\)-axis.

Find the two values of \(q\).
For the larger value of \(q\), solve \(f(x) = 0\).
Find the coordinates of the point of intersection of the two graphs.

IB Functions Problem 016

2026-05-21T11:55:29+00:00

Let \(f(x) = \ln(x +2)\), \(x > -2\) and \(g(x) = e^{x-4}\), \(x > 0\).

Find where the graph of \(f\) intersects the \(x\)-axis.
1. Find \(f(-1.999)\).
2. Find the range of \(f\).
Find the coordinates of the point of intersection of the graphs of \(f\) and \(g\).

IB Functions Problem 017

2026-05-21T11:55:28+00:00

The graph of a function \(f\) is shown in the figure below.

The point \(A(-1; 1)\) is on the graph and \(y=-1\) is a horizontal asymptote.

Let \(g(x) = f(x-1) + 2\). On the figure, sketch the graph of \(g\).
Write down the equation of the asymptote of \(g\).
Let \(A^\prime\) be the point on the graph of \(g\) corresponding to point \(A\). Write down the coordinates of \(A^\prime\).

IB Functions Problem 018

2026-05-21T11:55:27+00:00

Let the function be \(y = a\,\sin\,2x + c\), \(-180 \leq x \leq 180\), where \(x\) is measured in degrees.

The curve of the function is shown below.

Write down:
1. the period of this function;
2. the amplitude of this function.
Find the values of \(a\) and of \(c\).
Find the \(x\)-intercept of the curve with the negative part of the \(x\)-axis.

IB Functions Problem 019

2026-05-21T11:55:26+00:00

Let \(f(x) = \sin(e^x)\) for \(0 \leq x \leq 1.5\).

The following diagram shows the graph of \(f\).

Find the \(x\)-intercept of the graph of \(f\).

IB Functions Problem 020

2026-05-21T11:55:25+00:00

At an amusement park, a Ferris wheel with a diameter of \(111\) metres rotates at a constant speed.

The bottom of the wheel is \(k\) metres above the ground.

A seat starts at the bottom of the wheel.

The diagram is not to scale.

The wheel completes one rotation in 16 minutes.

After \(t\) minutes, the height of the seat above the ground is given by \(h(t) = 61.5 + a\,\cos\left(\frac{\pi}{2}t\right)\), for \(0 \leq t \leq 32\).

After \(8\) minutes, the seat is \(117\) m above the ground. Find \(k\).
Find the value of \(a\).
Find when the seat is \(30\) m above the ground for the third time.

IB Functions Problem 021

2026-05-21T11:55:24+00:00

Let \(f(x) = \frac{x - 5}{cx + 6}\) for \(x \ne -\frac{6}{c}\), \(c \ne 0\).

The line \(x = 3\) is a vertical asymptote of the graph of \(f\). Find the value of \(c\).
Write down the equation of the horizontal asymptote of the graph of \(f\).
The line \(y=k\), where \(k \in \mathbb{R}\), intersects the graph of \(\vert f(x) \vert\) at exactly one point. Find the possible values of \(k\).

IB Functions Problem 022

2026-05-21T11:55:23+00:00

Let \(f(x)=3x\), \(g(x)=2x - 5\) and \(h(x) = (f \circ g)(x)\).

Find \(h(x)\).
Find \(h^{-1}(x)\).

IB Functions Problem 023

2026-05-21T11:55:22+00:00

Let \(g(x) = \frac{1}{2}x\,\sin\,x\), for \(0 \leq x \leq 4\).

Sketch the graph of \(g\) on the axes below.

Hence find the value of \(x\) for which \(g(x) = -1\).

IB Functions Problem 024

2026-05-21T11:55:21+00:00

Consider \(f(x) = 2 - x^2\), for \(-2 \leq x \leq 2\), and \(g(x)= \sin\,e^x\), for \(-2 \leq x \leq 2\).

The graph of \(f(x)\) is given below.

On the diagram above, sketch the graph of \(g\).
Solve \(f(x) = g(x)\).
Write down the set of values of \(x\) such that \(f(x) > g(x)\).

IB Functions Problem 025

2026-05-21T11:55:20+00:00

The number of bacteria, \(n\), in a Petri dish after \(t\) minutes is given by \(n = 800e^{0.13t}\).

Find the value of \(n\) when \(t = 0\).
Find the rate of growth of \(n\) when \(t = 15\).
After \(k\) minutes, the rate of growth of \(n\) is greater than \(10\ 000\) bacteria per minute. Find the least value of \(k\), where \(k \in \mathbb{Z}\).