Topic 4: Functions and graphs

Make a table of values before you draw, and read answers where the graph meets the line you need.

Quick checks: multiple choice

Question 1Paper 1

The function

f

is defined by

f(x)

= 4x + 7

. What is the output when the input is

-3

Question 2Paper 1

f(x)

= 2x + 1

and

g(x)

= x^2

. Find

g(f(-3))

Question 3Paper 1

f(x)

= x + 4

and

g(x)

= 3x

. Which expression is

f(g(x))

Question 4Paper 1

A function assigns exactly one output to each input. Each set below lists couples (input, output). Which set is a function?

Question 5Paper 1

f(x)

= 4x - 7

. For which value of

x

f(x)

= 31

Question 6Paper 1

f(x)

= 2x^2 + kx + 1

, where

k

is a constant. Given that

f(3)

= 7

, find

k

Question 7Paper 1

f(x)

= 3x - 2

and

g(x)

= x + 4

. Solve

f(g(x))

= 4

Question 8Paper 1

The graph shows

y

= f(x)

, where

f(x)

= x^2 - 2x - 3

. Which values of

x

are the roots of

f(x)

= 0

Question 9Paper 1

The graph shows

y

= f(x)

, where

f(x)

= x^2 - 4x + 1

. What are the co-ordinates of its minimum point?

Question 10Paper 1

The graph shows

y

= f(x)

, where

f(x)

= x^2 - x - 2

, and the line

y

= 4

. Which values of

x

satisfy

f(x)

= 4

Question 11Paper 1

The graphs of

f(x)

= x^2 - 1

and

g(x)

= x + 1

are shown. For which values of

x

f(x)

= g(x)

Question 12Paper 1

The graph shows

y

= f(x)

, where

f(x)

= x^3 - x^2 - 4x + 4

. What are the roots of

f(x)

= 0

Question 13Paper 1

The graph shows

y

= f(x)

, where

f(x)

= x^3 - 3x + 2

. Its turning points are

(-1, 4)

and

(1, 0)

. For which values of

k

does

f(x)

= k

have three solutions?

Question 14Paper 1

f(x)

= 3(2^x)

. Find

f(4)

Question 15Paper 1

The graph of

y

= a(b^x)

passes through the points

(0, 5)

and

(1, 15)

, as shown. Which function is it?

Question 16Paper 1

The graph shows the straight line

y

= f(x)

through the points

(0, -2)

and

(3, 4)

. Which is

f(x)

Question 17Paper 1

The graph of a quadratic function

y

= f(x)

crosses the

x

-axis at

x

= -3

and

x

= 2

only. Which could be

f(x)

Question 18Paper 1

f(x)

= x^3 - 3x

. Find

f(-2)

, a value needed to draw the graph of

f

Question 19Paper 1

The number of bacteria in a dish

t

hours after it is set up is

N(t)

= 200(3^t)

. How many bacteria are there after 4 hours?

Question 20Paper 1

The graphs of

f(x)

= 2^x

and

g(x)

= x + 2

are shown. Which values of

x

, correct to one decimal place, satisfy

2^x

= x + 2

Paper-style questions: Section A, then Section B

Two functionsPaper 1, Section A style · 30 marks

The functions

f

and

g

are defined by

f(x)

= 3x - 2

and

g(x)

= x^2 + 1

, where

x \in \mathbb{R}

(a) Find

f(4)

. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(b) Find

g(-3)

. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

f(g(x))

in its simplest form. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(d) Solve

f(g(x))

= 28

. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

Two phone plansPaper 1, Section B style · 50 marks

Aoife is choosing a mobile phone plan. Plan A costs 15 euro a month plus 10 cent per minute of calls. Plan B has no monthly charge and costs 25 cent per minute of calls. For

x

minutes of calls in a month, the cost in euro of Plan A is

A(x)

= 15 + 0.1x

and the cost of Plan B is

B(x)

= 0.25x

(a) Find the cost of Plan A, in euro, for a month with 80 minutes of calls. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(b) Complete the table of monthly costs, in euro. [10]

Minutes, $x$	Plan A (euro)	Plan B (euro)
0	15	0
50
100
150
200

Type one value in each empty cell.

y

= A(x)

and

y

= B(x)

meet at one point. For how many minutes of calls do the two plans cost the same? [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(d) Find the cost, in euro, of either plan for that number of minutes. [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(e) Aoife expects to make 160 minutes of calls a month. How much, in euro, does she save each month by choosing the cheaper plan? [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(f) The monthly charge for Plan A rises to 18 euro; the cost per minute does not change. For how many minutes of calls do the two plans now cost the same? [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

Two growing culturesPaper 1, Section B style · 50 marks

A scientist grows two cultures of cells. The number of cells in culture N after

t

hours is

N(t)

= 300(2^t)

and the number in culture M is

M(t)

= 1200 + 600t

, for

0

\le t

\le 4

. The graphs of both functions are shown.

(a) How many cells are in culture N at the start, when

t

= 0

? [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(b) Complete the table of the number of cells. [10]

Time, $t$ (hours)	$N(t)$	$M(t)$
0	300	1200
1
2
3
4

Type one value in each empty cell.

(c) From the graph, estimate the time, in hours, when culture N has 2000 cells. Give your answer correct to one decimal place. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(d) From the graph, estimate the time, in hours, when the two cultures have the same number of cells. Give your answer correct to one decimal place. [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(e) By how many cells does culture N grow during the fourth hour, from

t

= 3

t

= 4

? [5]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.

(f) How many more cells are in culture N than in culture M when

t

= 4

? [10]

Type fractions as 3/8, powers as x^2, roots as sqrt(5), logs as ln(x) and complex numbers as 3+4i.