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**CAT Functions and Graphs**

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Functions and Graphs is one of the most difficult topics of CAT and XAT exam for beginners. But if you practice them it would be very very easy to understand them and master them. CAT has been giving difficult problems from this topic. A deep understanding of functions is required to solve such problems. We begin with a simple example to show you how to tackle these problems.

**CAT functions question 1 –**

Function f(x) = 2x + 3 and another function g(x) = 9x + 6. Find g(f(x)) – f(g(x)).

[1] 16

[2] 18

[3] 20

[4] 22

CAT functions solution:

g(f(x)) = g(2x+3) = 18x + 33

f(g(x)) = f(9x+6) = 18x + 15

so g(f(x)) – f(g(x)) = 18x + 33 – 18x + 15 = 18

Therefore, the correct option is [2].

**CAT functions question 2 –**

Function f(x) = 2x^2+2 and g(x) = x^3/f(x/2). Then find gof(3)

[1] 8×10^3

[2] 20^2×10^3

[3] 8×20^2×10^3

[4] None of these

CAT functions solution:

gof(3) = g[f(3)] = g[2*3^2+2] = g(20) = 20^3 /f(10) = 8000/202

Therefore, the correct option is [4].

**CAT functions question 3 –**

f(x + y) = f(x)f(y) for all x, y, f(4) = + 3 what is f(–8)?

[1] 1/3

[2] 1/9

[3] 9

[4] 6

CAT functions solution:

f(x + 0) = f(x) f(0)

f(0) = 1

f(4 + – 4) = f(0)

f(4 + – 4) = f(4) f(–4)

1 = +3 x f(–4)

f(-4) = 1/3

f(– 8) = f(– 4 + (– 4)) = f(– 4) f(– 4)

f(– 8) = 1/3 x 1/3 = 1/9

Answer choice (b).

**CAT functions questions 4 –**

Give the domain and range of the following functions:

f(x) = x2 + 1

g(x) = log(x + 1)

h(x) = 2x

f(x) = 1x+1

p(x) = |x + 1|

q(x) = [2x], where [x] gives the greatest integer less than or equal to x

CAT functions solution:

f(x) = x2 + 1

Domain = All real numbers (x can take any value)

Range [1, ∞). Minimum value of x2 is 0.

g(x) = log (x + 1)

Domain = Log of a negative number is not defined so (x + 1) > 0 or x > -1

Domain ( -1, ∞)

Range = (-∞, +∞)

Note: Log is one of those beautiful functions that is defined from a restricted domain to all real numbers. Log 0 is also not defined. Log is defined only for positive numbers

h(x) = 2x Domain – All real numbers.

Range = (0, ∞)

The exponent function is the mirror image of the log function.

f(x) = 1x+1

Domain = All real numbers except -1

Range = All real numbers except 0.

p(x) = |x + 1|

Domain = All real numbers

Range = [0, ∞) Modulus cannot be negative

q(x) = [2x], where [x] gives the greatest integer less than or equal to x

Domain = All real numbers

Range = All integers

The range is NOT the set of even numbers. [2x] can be odd. [2 * 0.6] = 1. It is very important to think fractions when you are substituting values.

**Basics concepts and scope of Functions**

• Questions on Functions (with Solutions). Several questions on functions are presented and their detailed solutions discussed.

• Linear Functions. A tutorial to explore the graphs, domains and ranges of linear functions.

• Square Root Functions. Square root functions of the form f(x) = a SQRT(x – c) + d and the characteristics of their graphs such as domain, range, x intercept, y intercept are explored interactively.

• Cube Root Functions. Cube root functions of the form f(x) = a (x – c) 1/3 + d and the properties of their graphs such as domain, range, x intercept, y intercept are explored interactively using an applet.

• Cubing Functions. Graphs of the cubing functions of the form f(x) = a (x – c) 3 + d as well as their properties such as domain, range, x intercept, y intercept are explored interactively using an applet.

• Graph, Domain and Range of Common Functions. A tutorial using a large window applet to explore the graphs, domains and ranges of some of the most common functions used in mathematics.

• Quadratic Functions (general form). Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an applet.

• Quadratic Functions(standard form). Quadratic functions in standard form f(x) = a(x – h) 2 + k and the properties of their graphs such as vertex and x and y intercepts are explored, interactively, using an applet.

• The Product of two Linear Functions Gives a Quadratic Function. This property is explored interactively using an applet.

• Even and Odd Functions. Graphical, using java applet, and analytical tutorials on even and odd functions.

• Periodic Functions. Use java applet to explore periodic functions.

• Definition of the Absolute Value. The definition and properties of the absolute value function are explored interactively using an applet. The properties of basic equations and inequalities with absolute value are included.

• Absolute Value Functions. Absolute value functions are explored, using an applet, by comparing the graphs of f(x) and h(x) = |f(x)|.

Exponential and Logarithmic Functions

• Exponential Functions. Exponential functions are explored, interactively, using an applet. The properties such as domain, range, horizontal asymptotes, x and y intercepts are also investigated. The conditions under which an exponential function increases or decreases are also investigated.

• Find Exponential Function Given its Graph.It is a tutorial that complements the above tutorial on exponential functions. A graph is generated and you are supposed to find a possible formula for the exponential function corresponding to the given graph.

• Logarithmic Functions. An interactive large screen applet is used to explore logarithmic functions and the properties of their graphs such domain, range, x and y intercepts and vertical asymptote.

• Gaussian Function. The Gaussian function is explored by changing its parameters.

• Logistics Function. The logistics function is explored by changing its parameters and observing its graph.

• Compare Exponential and Power Functions. Exponential and power functions are compared interactively, using an applet. The properties such as domain, range, x and y intercepts, intervals of increase and decrease of the graphs of the two types of functions are compared in this activity.

**Rational Functions**

• Rational Functions. Rational functions and the properties of their graphs such as domain, vertical and horizontal asymptotes, x and y intercepts are explored using an applet. The investigation of these functions is carried out by changing parameters included in the formula of the function.

• Rational Functions with Slant Asymptotes – Applet. Rational functions with slant asymptotes are explored interactively using an applet.

• Rational Functions with two Vertical Asymptotes – Applet. Rational functions with two vertical asymptotes are explored interactively using an applet.

Hyperbolic Functions

• Graphs of Hyperbolic Functions. The graphs and properties such as domain, range and asymptotes of the 6 hyperbolic functions: sinh(x), cosh(x), tanh(x), coth(x), sech(x) and csch(x) are explored using an applet.

Inverse of a Function and One to One Functions

• One-To-One functions. Explore the concept of one-to-one function using an applet. Several functions are explored graphically using the horizontal line test.

• Inverse Function Definition. The inverse function definition is explored using java applets. The conditions under which a function has an inverse are also explored.

• Inverse Functions. A large window applet helps you explore the inverse of one to one functions graphically. The exploration is carried out by changing parameters included in the functions.

Explore Other Functions

• Explore graphs of functions. This is an educational software that helps you explore concepts and mathematical objects by changing constants included in the expression of a function. The idea is to introduce constants ( up to 10) a, b, c, d, f, g, h, i, j and k into expressions of functions and change them manually to see the effects graphically then explore.