Identity Transformations of Expressions
p.1. Respective Values
Consider the following variable expressions:
$$ f(x)=x^2 - 4x + 20, g(x)=3x^2 - 10 $$
Evaluate them for x=2:
$$ f(2)=2^2 - 4 \cdot 2 + 20 = 16, g(2)=3 \cdot 2^2 - 10 = 2 $$
$$f(2) \neq g(2)$$
The numbers 16 and 2 are called the respective values of the expressions f(x)and g(x) for the same value x=2. In this case, the respective values are not equal. Now let us substitute x=3:
$f(3)=3^2 - 4 \cdot 3 + 20 = 17, g(3) = 3 \cdot 3^2 - 10 = 17$
$$f(3) = g(3)$$
The respective values are equal.
The respective values of two expressions containing the same set of variables are the numerical values these expressions return on substituting the same values for each of the variables.
The respective values can be:
- equal for certain values of the variables;
- equal for any of the allowable values of the variables;
- not equal for any of the allowable values of the variables.
p.2. Set of Allowable Values
The values of variables for which the algebraic expression has meaning are called allowable values for these variables..
The set of all allowable values for the variables is called the domain of the algebraic expression.
The domain restrictions are determined by the type of an expression:
- An integer expression has meaning for any values of its variables.
- • A fractional expression has no meaning for any of the variable values that makes the denominator equal to 0. For example, the expression $ \frac {1}{a-4}$ has no meaning for a=4.
- • An irrational expression has no meaning if the expression under an even root or under a fractional power is negative. For example, the expression $ \sqrt {a-1}$ has no meaning for any a < 1; the expression $a^{\frac 23}-b^{\frac 13}$ has no meaning for any a < 0 and b < 0.
p.3. Identities and Identity Transformations of Expressions
Expressions are called identical (or, identically equal) if their respective values are equal for any variable value in the domain.
An identity is a formula with two identically equal expressions connected by an equal sign.
By definition, an identity is an equality that is true for all allowable values of its variables.
Examples of Identities: $a + b = b + a, \frac {2a+2}{2} = a+1, x^2 - 1 = (x - 1)(x + 1)$
True numerical equalities are also considered identities.
Examples of Numerical Identities: $3^2 + 4^2 = 5^2, 1 + 3 + 5 + 7 = 4^2$
The difference between an identity and an equation is that an identity is true for all allowable values of the variables, while an equation is true for only one or several values from the domain.
For example, $x + 1 = \frac {2x+2}{2}$ is an identity that is true for any real $x \mathbb \in R$. Meanwhile, the expression $x^2 + 1 = 2$ is an equation that is true for $x = \pm 1$ only.
An identity transformation of an expression is the replacement of this expression by another, identically equal to it.
For example, reducing the fraction $ \frac {ac}{bc} = \frac ab $ is an identity transformation.
The following algorithms are used to prove (or disprove) identities.
Algorithm for Proving that Equality is Identity
1. Perform identity transformations of one or both parts of the equality.
2. Compare the algebraic expressions obtained on the left and on the right. If they are the same, then equality is an identity.
If the expressions are not the same, keep on doing the identity transformations or pass to the disproving algorithm.
Algorithm for Disproving that Equality is Identity
Find at least one value of the variable for which the respective values of the expressions on the left and on the right are not equal.
p.4. Examples of Problem Solving
Example 1. Prove the identity 3(x+1)-2(x-1)-x=5(x+1)-5x
Proof:
● Identity transformations of the left part:
3(x+1)-2(x-1)-x=3x+3-2x+2-x=(3x-2x-x)+(3+2)=5
Identity transformations of the right part:
5(x+1)-5x=5x+5-5x=(5x-5x)+5=5
We obtain: 5=5. The equality is an identity.
Q.E.D. ○
Example 2. Are the expressions 1-(1-(1-b)) and 1-b?
Solution:
Identity transformations of the left part:
1-(1-(1-b))=1-1+(1-b)=1-b
Thus: 1-b=1-b. The expressions are identical.
Answer: Yes
Example 3. Is the equality |x|+1=|x+1| an identity?
Solution:
One value of the variable is enough to disprove an identity.
Let us find the respective values of the left and right parts for x=-1.
|-1|+1=1+1=2,|-1+1|=0
2 ≠ 0
Therefore, the equality is not an identity.
Answer: No
Example 4. Is the equality |a+b|=|a|+|b| an identity?
Solution:
Let us find the respective values of the left and right parts for a=-1, b=1.
|-1+1|=0, |-1|+|1|=2
0 ≠ 2
Thus, the equality is not an identity.
Answer: No
