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• Prakash Adhikari Islington college, Kathmandu
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Warm up time..
1. What is the type of the function alongside?
2. What is the domain of the function?
3. What are ranges of the function?
4. What is the co-domain of the function?
Warm Up time..
1. What is the name of the function g?
2. Tell the domain and range of the function
3. Is it possible to find the inverse of this function, state with reason
Discussion
Function
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PREVIEW of Last LectureDefinition of Relation and FunctionDomain and Range of FunctionDifferent types of Functions Inverse of FunctionComposition of two Functions f and g
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Learning description…Various forms of functionsOne to one Function, Determination of one to one functionInverse of Function (Linear and Quadratic form)Domain and Range of function Graph of the FunctionsGraph of one to one function and its inverse
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The set of fingerprints is uniquely defined for every person.
Set of Islington member Set of fingerprint
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One to One FunctionLet’s Revise the definition:• One to one Function is defined as no two elements in
domain of the function has same image in Range• In other words, A function f is said to be a one to one
function if each element is domain has each image in range.
• A function f:A→B is an One to One if x=y whenever f(x) = f(y)
Remember
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Let's solve an example• Determine a function f:x→ x−33 is one- to- one or not.
We have f(x) = x-33f(x) = f(y)
⇔x−33 = y−33 ⇒x = y
• Test for another one example for f:x→ −7We have f(x) = −7
f(x) = f(y) −⇔ 7 = −7
⇒ x = y So the above two functions determines as one-to-one function
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Let's solve an example• Determine a function f:x→ x2-3x is one- to- one or not.
We have f(x) = x2-3xf(x) = f(y)
⇔ x2-3x = y2-3y ⇒x2 = y2-3y-3x
⇒x = ⇒x ≠ y
So the function doesn’t determine one-to-one function
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Graphically..,
• We can also test the given function is one to one or not by Horizontal Line test
• A function is one to one if a horizontal line intersects(cuts) the graph in only one spot(point)
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Graphs of FunctionsGraph 1 Graph 2
Graph 3 Graph 4
What about these graphs? Are all these graphs are one to one Function?13
Various forms of Functions
• Linear Form:
f(x)=ax+b
• Quadratic Form:
f(x) =ax2+bx+c
• Radical Form:
f(x) = or f(x) = or f(x) =
• Trigonometric Form:
y=sinx0
or y= tanp0
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Graph of Function• Work on paper:
– Determine whether the functions are one- to- one or not.• f:x→ 4x, xєR• f:x→ x(x-2), xєR
– The function f and g are defined as follows: f:x→x2-2x ; xєR• Find the set of values of x for which f(x)˃15• State, with a reason, whether f has an inverse or not
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Domain and Range of f: A→B•Domain
–Domain is all REAL NUMBERS.–Defining Domain:–The set of numbers x for which a function f(x) is defined is called –domain of the function.
•Range :–Range of f is set of all y values lies in set B–Range is also all REAL NUMBERS–The set of numbers y for which a function is defined as images if every x of domain is called Range of the function
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Domain and Range of f: A→B•How to find Domain and Range of the function??
Example :• Let a function be f: x→
Since the domain x is all real numbers, then let’s assign any real number in input x
Solve to obtain the range of interval
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Domain and Range of f: A→B
What about quadratic Function??We have an example:F:x→-x2-2x-4SolutionWriting f(x) in the form of a(x+b)2+c theny= -(x+1)2-3
Or, (x+1)2=-3-y
Since LHS is perfect square then (x+1)2>=0 So as -3-y >=0It means y<= -3
Range of interval is (-Infinity, -3)
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Domain and Range of f: A→B
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Find the domain of the following functions and also write in interval notation
Inverse of a function
• It is possible to find the inverse of a function only when the function is one to one.
• In Inverse function range of f(x) is changed into domain for f-1(x)
• Steps to find Inverse for linear function– Suppose f(x) as y– Interchange the value of x and y– Finally, obtain the value of y in terms of x, which is the
inverse of Linear Function.
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Inverse of a function• Method 1• Find the inverse, f: x→ 2x+5 x є RSolution,
Let, y= f(x)Or, y= 2x+5
Interchanging the value of x and yOr, x= 2y+5Or, x-5= 2yOr, y = Or, f-1(x) = Ans.
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Methods of finding Inverse of functions• Method 2( tricky way)• Find the Inverse of f: x→ 2x+5, x є R Break down the function as,
x → [double] → [add 5] →yTo find f-1 lets go backwards through the chain ½(x-5) ← [halve] ← [subtract] ←x• So f-1:x→ = Ans
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Inverse of a function • Inverse for linear function- f(x)=ax2+bx+c
Steps:
– Suppose f(x) as y
– Write the function as y= a(x+b)2
+c ( by completing the square)
– Interchange the value of x and y
– Finally, obtain the value of y in terms of x, which is the inverse of
Linear Function
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Inverse of Function• Example: Compute the inverse of: f(x)= x2+2x+6Solution, Let f(x)=y
Then, y= x2+2x+6 Or, y= (x2+2.x.1+12-12)+6 [ completing the square
rule]Or, y= [(x+1)2-1]+6Or y= (x+1)2+5 which is in the form of y=a(x+b)2+c
Now, to find f-1, interchanging x and y, X = (y+1)2+5 Or, x-5 =(y+1)2
Or, [ giving Sq. Root on both sides] Or y=-1
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Inverse of Function• Work on paper: Find the inverse of following functions
1. f:x→ x2-6x, xєR and x>= 02. f:x→ x2+x+6, xєR and x> 03. f:x→ -2x2+4x-7, xєR and x<14. f:x→ x2-2x+7, xєR
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Graph of Function
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Graph basics…
Quadrant IX>0, y>0
Quadrant IIX<0, y>0
Quadrant IIIX<0, y<0
Quadrant IVX>0, y<0
Origin (0,0)
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Graphing an equation in 2 variablesGraph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation.
How to Graph a function???1. Construct a table of values2. Graph enough solutions to recognize a pattern3. Connect the points with a line or curve
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Graph the function: f(x) = x + 1 or, y = x + 1Step1: Table of values of y=x+1 Step2: Plotting the point and Graphing
X Y Order pair
-3 -2 (-3,-2)
-2 -1 (-2,-1)
-1 0 (-1,0)
0 1 (0,1)
1 2 (1,2)
2 3 (2,3)
3 4 (3,4)
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Compare graphs with the graph f(x) = x. Graph the function f(x) = x + 3, then compare it to the another
function g(x) = x.
g(x) = xf(x) = x + 3
The graphs of g(x) and f(x) have the same slope of 1.
g(x) = x
x f(x)
-5 -5
-2 -2
0 0
1 1
3 3
x f(x)
-5 -2
-2 1
0 3
1 4
3 6
f(x) = x + 3
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Compare graphs with the graph f(x) = x. Graph the function h(x) = 2x, then compare it to the
another function f(x) = x.
x h(x)
-3 -6
-2 -4
0 0
2 4
3 6
f(x) = x
f(x) = xx f(x)
-5 -5
-2 -2
0 0
1 1
3 3
h(x) = 2xh(x) = 2x
The graphs of h(x) and f(x) both have a y-int of 0. The slope of h(x) is 2 and therefore is steeper than f(x) with a slope of 1. 31
Graphing Quadratic Function
y = ax2 + bx + c
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Graphing Quadratic Functions
The graph of a quadratic function is a parabola.
A parabola can open up or down.
If the parabola opens up, the lowest point is called the vertex.
If the parabola opens down, the vertex is the highest point.
y
x
Vertex
Vertex
Standard form of quadratic function is y.= ax2+bx+c
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y = ax2 + bx + c
The parabola will open down when the a value is negative.
The parabola will open up when the a value is positive.
Graphing Quadratic FunctionsStandard Form
a > 0
a < 0
y
x
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y
x
Graphing Quadratic FunctionsLine of Symmetry
Parabolas have a symmetric property to them.
If we draw a line down the middle of the parabola, we could fold the parabola in half.
The line of symmetry ALWAYS passes through the vertex.
Line of Semmetry
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Graphing Quadratic FunctionsFinding the Line of Symmetry and vertex
When a quadratic function is in standard form
The equation of the line of symmetry is
y = ax2 + bx + c,
2ba
x
(the opposite of b divided by the quantity of 2 times a).
We know the line of symmetry always goes through the vertex.
Thus, the line of symmetry gives us the x – coordinate of the vertex.
To find the y – coordinate of the vertex, we need to plug the x – value into the original equation.
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Graphing Quadratic Functions
There are 3 steps to graphing a parabola in standard form.
STEP 1: Find the line of symmetry
STEP 2: Find the vertex
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
MAKE A TABLE
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STEP 1: Find the line of symmetry
Let's Graph ONE!
y = 2x2 – 4x – 1
( )4
12 2 2
bx
a
-= = =
A Quadratic Function Standard Form
y
x
Thus the line of symmetry is x = 1
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STEP 2: Find the vertex
A Quadratic Function in Standard Formy
x
Thus the vertex is (1 ,–3).
Since the x – value of the vertex is given by the line of symmetry, we need to plug in x = 1 to find the y – value of the vertex.
= -3
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5
–1
STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve.
A Quadratic Function in Standard Formy
x
3
2
yx
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Graph of f(x) and f-1(x)• If f is one-to-one function, The graphs of y=f(x)
and y=f-1(x) are reflection of each other in the line y=x
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Line segment y=x
Graph of f(x) and f-1(x)
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Figure 1 Figure 2
Graph of Function• Work on paper: Sketch the graph of the
following functions.• Sketch the following functions:• f:x→ 4x, xєR• f:x→ X2, xєR and x≠0• Find the inverse function f:x→1-3x, xєR and sketch the graph
of y=f(x) and y= f-1(x)
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References / for exercise• Cambridge University Press ‘A Level Mathematics, Pure Mathematics 1’
page: 32 and 169• As Level Mathematics 9709- Past papers compilation
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