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pappoelarry
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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)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.
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.
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.
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.
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.
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.