Introduction to learn pre calculus test:

Calculus is the study of rates of change, measurement of changing quantities using symbolic notations. In precalculus identifies does not involved in calculus but explore topics that will be applied in calculus.

 

Calculus two types are:          

1)Differentiation

2) Integration.

Precalculus functions shown in below

1) Domain and Range, Composition.

2) Difference of Quotients

Differential calculus analyses are denoted as rates of change. Integral calculus measure the size of area, and volume calculus

 

Pre-calculus test problems:

 

Example for precalculus problem 1: Use inverse functions to solve equations.

Solve the following equation

Log ( x - 3) = 2

Solution to example 1:

  • Logarithmic functions and exponential functions are inverses of each other, we can write the following.

    A = Log (B) if and only B = 10
    A
  • Use the above property of logarithmic functions and exponential functions to rewrite the given equation as follows.

    x - 6 = 10
    2
  • Solve x.

    x = 106

Example for precalculus problem 2: Use inverse functions to find range of functions.
                              f(x) = 2 x / (x - 3)

Solution to example 2:

  •  Range of the one to one function is the domain of its inverse. Let us first prove that function f given above is a one to one function.  

    f(a) = f(b)

    2 a / (a - 3) = 2 b / (b - 3)
  • Multiply all conditions of the above equation by (a - 3) (b - 3) and simplify to obtain.

    2a (b - 3) = 2 b(a - 3)
  • multiply inside the brackets.

    2a b - 6 a = 2 ba - 6 b
  • Add - 2 a b to calculate both sides.

    a = b
  • So the given function is a one to one. let us find its inverse.

    y = 2 x / (x - 3)
  • Interchange x and y and solve for y.

    x = 2 y / (y - 3)

    y = 3x / (2 - x)
  • The inverse of function f is given by.

    f
    -1 (x) = 3x / (2 - x)

The domain of f-1 is set of all real values except x = 2. Since  the range of f is  the set of all real values except 2.

 

Pre calculus test problem:

Example for precalculus problem 3: Use inverse functions to solve equations. 

Solve the following equation

Log ( x - 4) = 3

Solution to example 3:

  • Logarithmic functions and exponential functions are inverses of each other, we can write the following.

    A = Log (B) if and only B = 10 A
  • Use the above property of logarithmic functions and exponential functions to rewrite the given equation as follows.

    x - 4 = 10 3
  • Solve for x to obtain.

    x = 1004