# Proving Equivalence of f(x) and g(x)

• MHB
• pappoelarry
In summary, we are considering the equivalence of two functions, f(x) and g(x). By simplifying each function and adding like terms, we see that f(x) = 3x^2 + 5x - 7 and g(x) = 3x^2 + 3x - 7. Since they are not equivalent, we cannot prove that they are the same.
pappoelarry
Consider the two functions f(x)=(x²+3x+10)+(2x²+2x-17) and g(x)=(4x²+4x+4)-(x²+x+11). If they are equivalent, prove they are; if they are not equivalent, prove they aren't.

The first function can be simplified by adding like terms: $$\displaystyle f(x) = 3x^2 + 5x - 7$$. Do the same for g(x). Do they come out the same?

-Dan

pappoelarry said:
Consider the two functions f(x)=(x²+3x+10)+(2x²+2x-17) and g(x)=(4x²+4x+4)-(x²+x+11). If they are equivalent, prove they are; if they are not equivalent, prove they aren't.
f(x)=(x²+3x+10)+(2x²+2x-17)= (x^2+ 2x^2)+ (3x+ 2x)+(10- 17)
Can you finish that ?

g(x)=(4x²+4x+4)-(x²+x+11)= (4x^2- x^2)+ (4x- x)+ (4- 11)
Can you finish that?

Are they the same?

## 1. How do you prove equivalence of two functions?

To prove equivalence of two functions, you need to show that they have the same output for every input. This can be done by using algebraic manipulations, graphing, or mathematical induction.

## 2. What is the difference between proving equivalence and proving equality of two functions?

Proving equivalence means showing that two functions have the same output for every input, while proving equality means showing that two functions are exactly the same. Two functions can be equivalent without being equal, but if they are equal, then they are also equivalent.

## 3. Can two functions be equivalent but have different forms?

Yes, two functions can be equivalent but have different forms. For example, the functions f(x) = x^2 and g(x) = x^2 + 0.5x - 0.25 are equivalent because they have the same output for every input, but they have different forms.

## 4. What are some common techniques used to prove equivalence of functions?

Some common techniques used to prove equivalence of functions include substitution, simplification, and manipulation of algebraic expressions, as well as using properties of functions such as symmetry, monotonicity, and periodicity.

## 5. Can you prove equivalence of two functions without using algebraic manipulations?

Yes, it is possible to prove equivalence of two functions without using algebraic manipulations. For example, you could use graphing to show that the two functions have the same shape and intersect at every point, or you could use mathematical induction to prove that the two functions have the same output for every input.

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