Function II

Functions: Domain and Range

By Mr Porter

-2 4

(1,-9)

axis

1

X-axis

Y-axis

y = x2 - 2x - 8

- 8

X-axis

Y-axis

y = mx + b

b

-bm

Definitions

Function:A function is a set of ordered pair in which no two ordered pairs have the same x-coordinate.

DomainThe domain of a function is the set of all x-coordinates of the ordered pairs.[the values of x for which a vertical line will cut the curve.]

Range

The range of a function is the set of all y-coordinates of the ordered pairs.[the values of y for which a horizontal line will cut the curve]

Note: Students need to be able to define the domain and range from the equation of a curve or function. It is encourage that student make sketches of each function, labeling each key feature.

Linear Functions

Any equation that can be written in the • General form ax + by + c = 0• Standard form y = mx + b

Sketching Linear Functions.Find the x-intercep at y = 0And the y-intercept at x = 0.

Examplesa) y = 3x + 6 b) 2x + 3y = 12

x-intercept at y = 0 0 = 3x + 6

x = -2y-intercept at x = 0

y = 6

Every vertical line will cut y = 3x + 6.

Every horizontal line will cut y = 3x + 6

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Domain : All x ∈ R , real numbers

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Range : All y ∈ R , real numbers

x-intercept at y = 0 2x = 12

x = 6y-intercept at x = 0

3y = 12y = 4-2

6

Y-axis

X-axis

y = 3x + 6

Y-axis

X-axis

4

6

Every vertical line will cut 2x+3y=12.

Every horizontal line will cut 2x+3y=12 6

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Domain : All x ∈ R , real numbers

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Range : All y ∈ R , real numbers

Special Lines

Vertical Lines: x = a- these are not functions, as the first element in any ordered pair is (a, y)

Equation of a vertical line is:i) x = aii) x - a = 0

Sketch (a,b)x

= a

X-axis

Y-axis

Domain: x = a

Range: all y in R

Examples

a) x = 4

b) x + 2 = 0

(4,5)

x =

4

X-axis

Y-axis

(-2,-6)

x =

-2

X-axis

Y-axis

Domain: x = 4

Range: all y in R

Domain: x = -2

Range: all y in R

Special Lines

Horizontal Lines: y = a- these are functions, as the first element in any ordered pair is (x, a)

Equation of a horizontal line is:i) y = aii) y - a = 0

Sketch (a,b) y = a

X-axis

Y-axis

Domain: all x in R

Range: y = b

Examples

a) y = 3

b) y + 6 = 0

(-5,3)y = 3

X-axis

Y-axis

(2,-6)y = -6

X-axis

Y-axis

Domain: all x in R

Range: y = 3

Domain: all x in R

Range: y = -6

Parabola: y = ax2 +bx + c

The five steps in sketching a parabola function:

1) If a is positive, the parabola is concave up.If a is negative, the parabola is concave down.

2) To find the y-intercept, put x = 0.3) To find the x-intercept, form a

quadratic and solve ax2 + bx + c = 0

* factorise* quadratic formula

4) Find the axis of symmetry by

5) Use the axis of symmetry x-value to find the y-value of the vertex, h

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x =−b2a

ExampleSketch y = x2 + 2x - 3, hence, state its domain and range.1) For y = ax2 + bx + c a = 1, b = +2, c = -3 . Concave-up a = 1

2) y-intercept at x = 0, y = -3

3) x-intercept at y = 0, (factorise )(x - 1)(x + 3) = 0x = +1 and x = - 3.

4) Axis of symmetry at = -1

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x =−b2a

=−(2)2(1)

5) y-value of vertex: y = (-1)2 +2(-1) - 3 y = -4

Domain: all x in RRange: y ≥ h for a > 0Range: y ≤ h for a < 0

Domain: all x in R Range: y ≥ -4

-3 1

(-1,-4)

-3

X-axis

Y-axis

-1

Parabola: y = ax2 +bx + c

The five steps in sketching a parabola function:

1) If a is positive, the parabola is concave up.If a is negative, the parabola is concave down.

2) To find the y-intercept, put x = 0.3) To find the x-intercept, form a

quadratic and solve ax2 + bx + c = 0

* factorise* quadratic formula

4) Find the axis of symmetry by

5) Use the axis of symmetry x-value to find the y-value of the vertex, h

€

x =−b2a

ExampleSketch y = –x2 + 4x - 5, hence, state its domain and range.1) For y = ax2 + bx + c, a = -1, b = +4, c = -5. Concave-down a = -1

2) y-intercept at x = 0, y = -5

3) x-intercept at y = 0, NO zeros by Quadratic formula.

4) Axis of symmetry at = +2

€

x =−b2a

=−(4)2(−1)

5) y-value of vertex: y = -(2)2 +4(2) - 5 y = -1

Domain: all x in R Range: y ≤ -1

(2,-1)-5

X-axis

Y-axis

2

Domain: all x in RRange: y ≥ h for a > 0Range: y ≤ h for a < 0

Worked Example 1: Your task is to plot the key features of the given parabola, sketch the parabola, then state clearly its domain and range.

Sketch the parabola y = x2 - 2x - 8, hence state clearly its domain and range.

The five steps in sketching a parabola function:1) If a is positive, the parabola is concave up.

If a is negative, the parabola is concave down. 2) To find the y-intercept,

put x = 0.3) To find the x-intercept, form a quadratic and solve

ax2 + bx + c = 0* factorise* quadratic formula

•Find the axis of symmetry by

5) Use the axis of symmetry x-value to find the y-value of the vertex, h €

x =−b

2a

Step 1: Determine concavity: Up or Down?

For the parabola of the form y = ax2 + bx + c a = 1 => concave up

Step 2: Determine y-intercept.

Let x = 0, y = -8

Step 3: Determine x-intercept.

Solve: x2 - 2x - 8 = 0

Factorise : (x - 4)(x + 2) = 0 ==> x = 4 or x = -2.

Step 4: Determine axis of symmetry.

€

x =−b2a

€

=−(−2)2(1)

=1

Step 5: Determine maximum or minimum y-value (vertex).

Substitute the value x = 1 into y = x2 - 2x - 8.

y = (1)2 - 2(1) - 8 = -9 Vertex at (1, -9) Domain all x in RRange y ≥ -9

-2 4

(1,-9)

axis

1

X-axis

Y-axis

y = x2 - 2x - 8

- 8

Worked Example 2: Your task is to plot the key features of the given parabola, sketch the parabola, then state clearly its domain and range.

Sketch the parabola f(x) = 15 - 2x - x2, hence state clearly its domain and range.

Step 1: Determine concavity: Up or Down?

For the parabola of the form f(x) = ax2 + bx + c a = -1 => concave down

Step 2: Determine y-intercept.

Let x = 0, f(x) = +15

Step 3: Determine x-intercept.

Solve: 15 - 2x - x2 = 0

Factorise : (3 - x)(x + 5) = 0 ==> x = 3 or x = -5.

Step 4: Determine axis of symmetry.

€

x =−b2a

€

=−(−2)2(−1)

= −1

Step 5: Determine maximum or minimum y-value (vertex).

Substitute the value x = -1 into y = 15 - 2x - x2.

y = 15 - 2(-1) - (-1)2 = 16 Vertex at (1, 16) Domain: all x in RRange: y ≤ 16

The five steps in sketching a parabola function:1) If a is positive, the parabola is concave up.

If a is negative, the parabola is concave down. 2) To find the y-intercept,

put x = 0.3) To find the x-intercept, form a quadratic and solve

ax2 + bx + c = 0* factorise* quadratic formula

•Find the axis of symmetry by

5) Use the axis of symmetry x-value to find the y-value of the vertex, h €

x =−b

2a

-5 3

(1,16)

axis

-1

X-axis

Y-axis

f(x)=15 - 2x - x2

15

Exercise: For each of the following functions:a) sketch the curveb) sate the largest possible domain and range of the function.

(i) f(x) = 5 - 2x (ii) h(x) = 2x2 + 7x - 15

(iv) g(x) = 5x + 4 (iii) h(x) = x2 + 2x + 5

X-axis

Y-axis

f(x) = 5 - 2x

5

21/2

Domain: All x in RRange: All y in R

-5 11/2 -13/4

Y-axis

X-axis

h(x) = 2x2 + 7x - 15

-15

(-13/4 ,-211/8 )

Domain: All x in RRange: All y ≥ -211/8

NO x-intercepts.(try quadratic formula?)

-1

Y-axis

X-axis

h(x) = x2 + 2x + 5

5

(-1 ,4)

Domain: All x in RRange: All y ≥ 4

X-axis

Y-axis

g(x) = 5x + 4

4

-4/5

Domain: All x in RRange: All y in R