Concept of a Function

Consider y = 2x. x y 1 2 2 4 3 6 … … Each value of x determines exactly one value of y. y is a function of x.

Consider the function y = x + 1. 5 6 y = x + 1 Input x Output y y = 5 + 1

Consider the function y = x + 1. 10 11 y = x + 1 Input x Output y y = 10 + 1 Independent variable Dependent variable The value of y depends on the value of x.

If y2 = 9x, is y a function of x? ∵ When x = 1, y = 3 or –3. i.e. There is more than one value of y for x = 1. ∴ y is not a function of x.

Each value of x gives exactly one value of y. Follow-up question Determine whether (where x ≠ 0) is a function of x. When x = 1, ; When x = 2, ; When x = 3, , etc. Each value of x gives exactly one value of y. ∴ y is a function of x.

Consider another function y = x2, where x = 1, 2, 3. Independent variable x 1 Collection of values that x can take is called the domain of the function. 2 3 domain

Consider another function y = x2, where x = 1, 2, 3. Independent variable Dependent variable x y Collection of values that y can take is called the range of the function. 1 1 2 4 3 9 domain range

Consider another function y = x2, where x = 1, 2, 3. Independent variable Dependent variable x y Collection of values that must include all possible values of y is called the co-domain of the function. 1 1 2 4 3 9 domain range co-domain

Follow-up question Consider the function y = 2x, where x = 1, 2, 3, 4, … Find its domain and range, and state one of its possible co-domains. Domain: collection of 1, 2, 3, 4, … Range: collection of values that y can take

Follow-up question Consider the function y = 2x, where x = 1, 2, 3, 4, … Find its domain and range, and state one of its possible co-domains. Domain: collection of 1, 2, 3, 4, … Range: collection of 2, 4, 6, 8, … Co-domain: all even numbers

Follow-up question Consider the function y = 2x, where x = 1, 2, 3, 4, … Find its domain and range, and state one of its possible co-domains. Domain: collection of 1, 2, 3, 4, … Range: collection of 2, 4, 6, 8, … Co-domain: 2, 4, 6, 8, … (or other possible answers like all positive numbers)

Different Representations of Functions In how many ways can we represent a function? 1. Algebraic Representation A table showing the relationship between x and y Express y in terms of x. e.g. y = 2x 2. Tabular Representation x 1 2 3 y 4 6

3. Graphical Representation x 1 2 3 6 4 2 y Plot the corresponding values of x and y on a coordinate plane. y = 2x From the graph, there is exactly one value of y for each value of x.

Follow-up question For each of the following graphs, determine whether y is a function of x, where 4  x  8 and y can be any real numbers. (a) y x ∵ Any vertical line intersects the graph at only one point. i.e. For any value of x where 4  x  8, there is only one corresponding value of y. 4 8 vertical lines ∴ y is a function of x.

Follow-up question For each of the following graphs, determine whether y is a function of x, where 4  x  8 and y can be any real numbers. (b) y x ∵ The vertical line on the left intersects the graph at two points. i.e. For a certain value of x where 4  x  8, there are more than one corresponding value of y. 4 8 vertical line ∴ y is not a function of x.

Notation of a Function

Apart from using y, we can use different notations to denote different functions. For example, y = 10 – x f(x) = 10 – x y = 2a – 1 g(a) = 2a – 1 3 2 t y = 3 ) ( 2 t H =

Find the values of the function when (a) x = 3, (b) x = –3. Consider f(x) = 10 – x. Find the values of the function when (a) x = 3, (b) x = –3. (a) When x = 3, f(3) = 10 – 3  When x = 3, the value of the function is f(3). = 7 (b) When x = –3, f(–3) =10 – (–3),  When x = –3, the value of the function is f(–3). = 13

Follow-up question If f(x) = ax + 8 and f(–2) = 2, find the value of a. ∵ f(–2) = 2 ∴ a(–2) + 8 = 2 –2a + 8 = 2 3 = a

In general, for a function y = f(x), (i) f(a) + f(b) ≠ f(a + b) (ii) f(a) – f(b) ≠ f(a – b) (iii) f(a)  f(b) ≠ f(ab) (iv) (v) kf(a) ≠ f(ka) where a, b and k are constants.

Some Common Functions and their Graphs – Constant Functions & Linear Functions

What is a constant function? Constant Functions What is a constant function? A function in the form y = c or f(x) = c, where c is a constant, is called a constant function.

Constant Functions What is a constant function? For example, y = 1 and f(x) = –2 are constant functions.

Consider the constant function y = 1. x –2 2 y 1 2 y 1 ◄ The value of y is always equal to 1, for any value of x. y x (–2, ) 1 (0, ) 1 (2, ) 1 y = 1 y-coordinates of all the points on the line are equal to 1 A straight line parallel to the x-axis

Graph of y = c O y x c 0 O y x c= 0 O y x c 0 (0, c) (0, c) (0, c) The graph of a constant function is a horizontal line passing through (0, c).

Follow-up question The figure shows the graph of a constant function. Write down the algebraic representation of the function. x y 2 1 1 2 3 4 2 4 ∵ The graph is a horizontal line which cuts the y-axis at (0, –3). (0, –3) ∴ The required function is y = –3. y = f(x)

Linear Functions A function in the form y = ax + b or f(x) = ax + b, where a and b are constants and a  0, is called a linear function of x. Example: (i) y = 3x – 1 linear functions of x (ii) f(x) = –2x + 5

The graph cuts the x-axis at P. Graph of y = ax + b O y x y = ax + b The graph cuts the x-axis at P. P

The graph cuts the y-axis at Q. Graph of y = ax + b O y x y = ax + b Q The graph cuts the y-axis at Q. x-intercept a b -

Graph of y = ax + b x-intercept y-intercept The graph of a linear function is a straight line. It has only one x-intercept and one y-intercept.

Graph of y = ax + b a 0 a 0 O y x O y x The value of y increases as x increases. The value of y decreases as x increases.

Follow-up question The figure shows the graph of the linear function y = f(x). Write down the x-intercept and the y-intercept of the graph. x y 2 1 1 2 3 4 4 2 2 4 ∵ The graph cuts the x-axis and the y-axis at (2, 0) and (0, 3) respectively. y = f(x) (0, 3) (2, 0) ∴ x-intercept = 2 and y-intercept = 3

Some Common Functions and their Graphs – Quadratic Functions

quadratic functions of x A function in the form y = ax2 + bx + c or f(x) = ax2 + bx + c, where a, b and c are constants and a  0, is called a quadratic function of x. Example: (i) y = –2x2 + x quadratic functions of x (ii) f(x) = (x + 3)2 + 5

Graph of y = ax2 + bx + c (a > 0) axis of symmetry The graph has reflectional symmetry about . y x opens upwards for a > 0 a vertical line c The graph and its axis of symmetry intersect at . y-intercept = c a point x-intercepts vertex (It is the minimum point of the graph.) minimum value of y

Graph of y = ax2 + bx + c (a < 0) axis of symmetry y x vertex (It is the maximum point of the graph.) maximum value of y opens downwards for a < 0 x-intercepts c y-intercept = c

Coordinates of the vertex = (–1, 0) Can you find the axis of symmetry and the coordinates of the vertex of the graph of y = x2 + 2x + 1? axis of symmetry x y 4 3 2 1 1 2 8 6 4 2 y = x2 + 2x + 1 Axis of symmetry: x = –1 vertex Coordinates of the vertex = (–1, 0)

Follow-up question Consider the graph of y = (x + 1)(5 – 2x). (a) Determine the direction of opening. (b) Find the x-intercept(s) and the y-intercept of the graph. y = (x + 1)(5 – 2x) (a) y = 5x + 5 – 2x2 – 2x ∴ y = –2x2 + 3x + 5 ∵ Coefficient of x2 = –2  0 ∴ The graph opens downwards.

+ (b) ∵ y = –2x2 + 3x + 5 ∴ The y-intercept of the graph is 5. The x-intercepts of the graph are the roots of (x + 1)(5 – 2x) = 0. ) 2 5 )( 1 ( = - + x 2 5 or 1 = - + x 2 5 or 1 = - x ∴ The x-intercepts of the graph are –1 and . 2 5

Quadratic Functions in the Form y = a(x – h)2 + k

Can we obtain the features of the graph of a quadratic function without plotting its graph? Yes. For the function y = a(x – h)2 + k, we can obtain the axis of symmetry and the coordinates of the vertex directly.

Let us study quadratic functions in the form y = a(x – h)2 + k. Can we obtain the features of the graph of a quadratic function without plotting its graph? Let us study quadratic functions in the form y = a(x – h)2 + k.

y = a(x – h)2 + k Case 1 : a > 0 a > 0 and (x – h)2  0 O y x a > 0 and (x – h)2  0 a(x – h)2  0 a(x – h)2 + k  k

The minimum value of y is k when x = h. y = a(x – h)2 + k axis of symmetry: x = h Case 1 : a > 0 O y x a > 0 and (x – h)2  0 a(x – h)2  0 y  k minimum value of y The minimum value of y is k when x = h. k vertex: (h, k)

y = a(x – h)2 + k Case 2 : a < 0 a < 0 and (x – h)2  0 O y x a < 0 and (x – h)2  0 a(x – h)2  0 a(x – h)2 + k  k

The maximum value of y is k when x = h. y = a(x – h)2 + k vertex: (h, k) Case 2 : a < 0 O y x a < 0 and (x – h)2  0 maximum value of y k a(x – h)2  0 y  k The maximum value of y is k when x = h. axis of symmetry: x = h

Maximum or minimum value Features of the function y = a(x – h)2 + k and its graph a > 0 a < 0 Direction of opening Axis of symmetry Vertex Maximum or minimum value upwards downwards x = h (h, k) (min. point) (max. point) min. value = k max. value = k

∵ Coefficient of x2 = 1 > 0 Can you find the minimum value of the function y = (x + 5)2 + 7 and the axis of symmetry of its graph? y = a(x – h)2 + k y = (x + 5)2 + 7 = [x – (–5)]2 + 7 ◄ a = 1, h = –5, k = 7 ∵ Coefficient of x2 = 1 > 0 min. value = k ∴ The minimum value of y is 7. ∴ The axis of symmetry is x = –5. x = h

Follow-up question For the function y = 9 – 2(x + 1)2, find (a) its maximum or minimum value, (b) the coordinates of the vertex. (a) y = 9 – 2(x + 1)2 = –2(x + 1)2 + 9 For y = a(x – h)2 + k, coordinates of the vertex: (h, k) = –2[x – (–1)]2 + 9 ∵ Coefficient of x2 = –2 < 0 ∴ The maximum value of y is 9. (b) Coordinates of the vertex = (–1, 9)

Using the Algebraic Method to Find Optimum Values of Quadratic Functions

Forming Perfect Squares Consider x2 + 6x. By adding , we have x2 + 6x = x2 + 2(3)x + 32 ◄ x2 + 2mx + m2 ≡ (x + m)2 = (x + 3)2 a perfect square This process is called completing the square.

Completing the square To complete the squares for x2 + kx and x2 – kx, add to each expression. Then

How to find the optimum value of the quadratic function y = ax2 + bx + c? Convert y = ax2 + bx + c to the form y = a(x – h)2 + k by completing the square first.

Optimum values of quadratic functions y = ax2 + bx + c y = a(x – h)2 + k completing the square b 2a +  2

Optimum values of quadratic functions y = ax2 + bx + c y = a(x – h)2 + k completing the square If a > 0, then the minimum value of y is k when x = h. If a < 0, then the maximum value of y is k when x = h.

Can you find the optimum value of the function y = x2 + 2x + 2? Complete the square for x2 + 2x. 2 ø ö ç è æ Add , i.e. 12. = x2 + 2x + 12 – 12 + 2 = (x2 + 2x + 1) – 1 + 2 = (x + 1)2 + 1 ∵ Coefficient of x2 = 1 > 0 ∴ The minimum value of y = x2 + 2x + 2 is 1.

Follow-up question Find the optimum value of the quadratic function y = –4x2 + 24x – 15, and the corresponding value of x. y = –4x2 + 24x – 15 = –4(x2 – 6x) – 15 Complete the square for x2 – 6x. 2 6 ø ö ç è æ Add , i.e. 32. = –4(x2 – 6x + 32 – 32) – 15 = –4(x2 – 6x + 9) + 36 – 15 = –4(x – 3)2 + 21 ∵ Coefficient of x2 = –4 < 0 ∴ The maximum value of y = –4x2 + 24x – 15 is 21 when x = 3.