Composition of Functions End of block exercises

Exercise (1)

Exercise (2)

Exercise (3)

Exercise (6)

Exercise (7)

Exercise (8)

Inverse Functions Exercise (1) Find the inverse of each of the following functions

y= f (x) is called the Cartesian form, An alternative representation is to write expressions for y and x in terms of a third variable known as a parameter, such as y (t), x (t). Parametric Representation of a function

End of block exercises 1) Consider the parametric equation A) Plot a graph of this function B) Find an explicit expression for y in terms of x.

Parametric Representation of a function 2) Given the parametric equations Plot a graph of y against x.

Parametric Representation of a function 3) Obtain the Cartesian equation of the function defined parametrically by

Parametric Representation of a function 4) Plot a graph of the function define by

Describing Functions Continuous and Discontinuous functions The limit of function

Describing Functions Exercises 2) Study graphs of the functions Are these continuous functions?

Describing Functions 3) Study graphs of Are these continuous functions? 4)Draw a graph of the function (see a book for limits).

Periodic functions Any function that has a definite pattern repeated at regular intervals is said to be periodic, ( T period).

Odd and even functions Even function: any function that is symmetrical about the vertical axis. An even function is such that for all x.

Odd function: any function that possesses rotational symmetry- that is, the graph on the right can be obtained by rotating the curve on the left through 180º about the origin. An odd function is such that Odd and even functions

Exercises: 1) Classify the following as odd, even or neither. If necessary sketch a graph to help you decide

End of block exercises 1) Sketch a graph of the function

End of block exercises 2) A function is periodic with period 2 and is even. Sketch a possible form of this function. 3) A function is periodic with period 1 and is odd. Sketch a possible form of this function.

The straight line All linear functions can be written in the form f(x)=ax+b Or y=ax+b

The straight line Exercise: State which of the following functions will have straight line graphs:

In the linear function y= ax+b a is the gradient of the graph and b is its vertical intercept, The straight line

Exercise: For each of the following, identify the gradient and vertical intercept:

The straight line The gradient of the line joining A(x1,y1) and B(x2,y2) is given by

Exercise: 1) Calculate the gradient of the line joining (1,0) and (15,-3). 2) Calculate the gradient of the line joining (10,-3) and (15,-3) The straight line

The line passing through points A(x1,y1) and B(x2,y2) is given by

The straight line The distance between points A(x1,y1), B(x2,y2) is given by

Exercise: 1) Find the equation of the line joining (1,5) and (-9,2). 2) Find the gradient and vertical intercept of the line joining (8,1) and (-2,-3). 3) Find the distance between the points (4,5) and (-17,1). The straight line

End of block exercises: 1) State the gradient and vertical intercept of

3) Find the equation of the line that passes through A(0,3) and B(11,-1) 4) Find the gradient of the line that passes through A(-9,1) and B(2,16). 5) Find the distance between the points with coordinates (9,1) and (12,1). 6) Find the distance between the points with coordinates (19,-2) and (-12,1). The straight line

7) The pointsA(x1,y1) lies on the line y=-2x+3. If the value of x1 is increased by 7, what is the resulting change in the value of y1. 8) Find the equation of the line passing through A(2,-1) and B(5,8). The straight line

Common engineering functions Polynomial functions: A polynomial expressions has the form Where n is anon-negative integer.

Common engineering functions A polynomial function has the form

Common engineering functions Rational functions has the form Where P and Q are polynomial expressions: P is called the numerator and Q is called the denominator. The Poles of a rational function are any values that makes the denominator zero.

Common engineering functions The modulus function is defined as

Common engineering functions The Unit step function is defined as follows

Common engineering functions The delta function, or unit impulse function

Common engineering functions Exercises:

Common engineering functions 8) The signum function is defined as form below. a) Sketch a graph of this function b) Is this function discontinuous or continuous? c) Is this function odd, even, or neither? d) Is this function periodic? e) Is this function many-to-one or one-to-one?

Common engineering functions 9) a) Sketch a graph of the function b) State the position of any discontinuous.

Common engineering functions 10) The ramp function is defined as below a) Sketch a graph of this function b) State the position of any discontinuous. c) Find

Common engineering functions 11) Sketch a graph of 12) Sketch a graph of

Common engineering functions 13) State the poles of the following rational functions:

Common engineering functions 14) Consider the function

Common engineering functions 15) Find the inverse of the function

Common engineering functions 16)

Common engineering functions 17) State the rule that describes the function

Common engineering functions 18) Write a formula for the function given by the rule ‘subtract the cube of the input from the square of the input’. 19) State the domain and range of the function

Common engineering functions 20) The maximal domain of a function is the largest possible domain that can be defined for that function. Find the maximal domain of the function

Common engineering functions 21) Find the inverse of the function 22) Find the equation of the straight line passing through (-1,4) and (-4,1). Does the line pass through(-2,3)?

Common engineering functions Electrical Engineering and Electronics- Reactance of a capacitor. The reactance of a capacitor is its resistance to the passage of alternating current. Reactance, X, measured in ohms, is given by Where f is the frequency of the current in Hertz and C is the capacitance, measured in farads. Note that X is a function of f. Calculate the reactance when the frequency is 50 Hertz and the capacitance is Farads.

26) Extension of a spring. A spring has a natural length of 90cm. When a 1.5kg mass is suspended from the spring, the length extends to 115cm.Calculate the length when a 2.5kg mass is suspended from the spring. Common engineering functions

27) A curve is defined parametrically by Obtain y explicitly in terms of x.