If f(x) = 3x2 – 2x + 4 and g(x) = 2 – 3x, then find f(x)g(x) for x = -1

Mathematics is not only about numbers but it is about dealing with different calculations involving numbers and variables. This is what basically known as Algebra. Algebra is defined as the representation of calculations involving mathematical expressions that consist of numbers, operators, and variables. Numbers can be from 0 to 9, operators are the mathematical operators like +, -, ×, ÷, exponents, etc, variables like x, y, z, etc.

Exponents and Powers

Exponents and powers are the basic operators used in mathematical calculations, exponents are used to simplifying the complex calculations involving multiple self multiplications, self multiplications are basically numbers multiplied by themselves. For example, 7 × 7 × 7 × 7 × 7, can be simply written as 7⁵. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 11³, here, 11 is the base value and 3 is the exponent or power of 11. The value of 11³ is 1331.

Exponent is defined as the power given to a number, the number of times it is multiplied by itself. If an expression is written as cx^y where c is a constant, c will be the coefficient, x is the base and y is the exponent. If a number say p, is multiplied n times, n will be the exponent of p. It will be written as

p × p × p × p … n times = pⁿ

Functions

A function can be defined as the set of rules relating to a given set of inputs that provide some possible outputs. Only those expressions are denoted as functions in which there is one output for one input. Can there be two inputs for the same output? Yes. However, there cannot be two outputs for a single input.

Functions can be represented as f(x), g(x), h(x), etc. Here, f(x) is the output for a given value of input in the polynomial. For example, the value of f(x) for x = -2 in the function f(x) = 2x + 20 will be 16. It is obtained by placing the value of x in the expression and solving it.

Composite functions

Composite functions are obtained by putting one function in another function. It can be said that a composite function is obtained by solving a function in the function. For instance f{g(x)} is a composite function. Here, f(x’) is the final function value where x’ is another function known as g(x). Therefore, first, it is important to solve g(x) and then solve f(x’) with the help of g(x).

Multiplying and Dividing functions

In order to multiply or divide two functions, the first requirement is to understand that multiplication and division are basic mathematical multiplication and division. Just the way numbers are multiplied or divided, similarly, polynomials are multiplied and divided. They can be represented as f(x).g(x) for multiplication and f(x)/g(x) for division.

If f(x) = 3x² – 2x + 4 and g(x) = 2 – 3x, then find f(x)g(x) for x = -1

Solution:

In order to find the multiplication of both the functions, it can be either done by first multiplying the expressions and then putting the value of x or first put the value of x in separate functions and then multiply them. Since the functions are big and first multiplying will take time. It is smart to use the latter method.

f(x) = 3x² – 2x + 4

For x = -1, f(-1) = 3(-1)² – 2(-1) + 4

= 3 + 2 + 4 

= 9

g(x) = 2 – 3x

For x = -1, g(-1) = 2 – 3(-1)

= 2 + 3

= 5

f(x)g(x) for x = -1, 

f(-1)g(-1) = 9 × 5

= 45

Hence, the value obtained is 45.

Similar Problems

Question 1: Let f(x) = x + 7 and g(x) = x² – 4. Evaluate the product function f(x)g(x).

Solution:

To find f(x)g(x), simply multiply and then simplify the expression obtained.

f(x)g(x) = (x + 7).(x² – 4) 

= x³ – 4x + 7x² – 28

= x³ + 7x² – 4x – 28

Question 2: Let f(x) = x² – 49 and g(x) = x + 7. Evaluate the product function f(x)/g(x).

Solution:

To find f(x)/g(x), simply divide and then simplify the expression obtained.

f(x)/g(x) = (x² – 49) / (x + 7)

= (x² – 7²) / (x + 7)

Using identity, x² – y² = (x + y)(x – y)

= (x + 7)(x – 7) / (x + 7)

= x – 7

Question 3: Let f(x) = 2x³ + 4 and g(x) = 5 + x. Evaluate the product function f(x)g(x) for x = 1.

Solution:

In order to find the multiplication of both the functions, it can be either done by first multiplying the expressions and then putting the value of x or first put the value of x in separate functions and then multiply them. Since the functions are big and first multiplying will take time. It is smart to use the latter method.

f(x) = 2x³ + 4

For x = 1, f(1) = 2(1)³ + 4

= 2 + 4

= 6

g(x) = 5 + x

For x = 1, g(1) = 5 + 1

= 6

f(x)g(x) for x = 1,

f(1)g(1) = 6 × 6

= 36

Hence, the value obtained is 36.