Definition, domain and range, graphical representation
Monomial
A monomial function is a function of the following form:
f(x)=ax^n
a is a real number and n is a whole number.
The following functions are monomials:
- f(x)=2x^2
- g(x)=x
- h(x)=-17x^{101}
Degree of a monomial
Let a be a real number and n be a whole number. In the monomial ax^n, the exponent n is referred to as the degree of the monomial.
Let f be the monomial such that f\left(x\right)=3x^5. The degree of f is 5.
Every constant function is by definition a monomial function. If b is real number, then:
b=bx^0
The domain of a monomial function is all real numbers.
Consider the following function:
f(x)=2x^3
The domain of f(x) is all real numbers.
The range of a monomial function depends on whether or not the exponent is even or odd. Let f(x)=ax^n be a monomial function and assume that a>0. Then the range of f(x) is:
- All real numbers if n is odd.
- All nonnegative numbers if n is even.
Consider the following function:
f(x)=2x^3
The range of f(x) is all real numbers.
Consider the following function:
f(x)=2x^2
The range of f(x) is [0,\infty).
Let f(x)=ax^n be a monomial function, the graph of f(x) will resemble the following:
Properties and basic operations
If a and b are real numbers, then:
ax^n+bx^n=(a+b)x^n
ax^n-bx^n=(a-b)x^n
Therefore the sum of and difference of two monomials of the same degree is a monomial.
3x^2+17x^2=19x^2
3x^2-17x^2=-14x^2
The sum of two monomials of different degrees is not a monomial.
For example, the sum of the monomials x^2+7x^3 does not simplify to a monomial.
If a and b are real numbers, then:
ax^n\cdot bx^m=abx^{n+m}
Therefore the product of any two monomials is a monomial.
The product of the monomials 7x^3 and 3x^5 is:
7x^3\cdot 3x^5=21x^{3+5}=21x^{8}
When multiplying two monomials, the exponent of the new monomial is found by adding the previous exponents, not by multiplying the exponents.
Equations with monomials
Let a be a real number and then consider the following equation:
x^n=a
If n is an odd number, then the equation has a unique solution denoted:
x=a^{\frac{1}{n}}
Let's consider the equation x^3=27.
3 is odd so the equation has a unique solution.
-3 is the unique number such that (-3)^{3}=-27.
We denote:
(-27)^{1/3}=-3
Let a be a real number and then consider the following equation:
x^n=a
If n is an even number then the equation has a two solutions:
- One positive solution, which is denoted by a^{\frac{1}{n}}.
- The second solution: -a^{\frac{1}{n}}.
Let's consider the equation x^2=16.
2 is even so the equation has two solutions.
4 is the unique positive number such that 4^2=16.
We denote:
16^{1/2}=4
The number -4 also satisfies (-4)^2=16.
The equation has two solutions : 4 and -4.
Monomial equation
A monomial equation is an equation of the form:
ax^n=b
a and b are real numbers.
Consider the following monomial equation:
3x^2=12
The monomial equation is solved for by dividing by 3 and then applying the previous rules from above:
3x^2=12
x^2=4
x=2 or x=-2
Therefore the monomial equation has two solutions:
- x=2
- x=-2
Consider the following monomial equation:
-3x^2=12
It has no real solutions. This is shown as follows:
-3x^2=12
x^2=-4
The equation has no solution because no real number raised to an even power is negative.