Let f(x) = 5x^2 - 2x + 3 and g(x) = 4x^2 + 7x - 5. Find f + g and f - g.

Understand the Problem

The question provides mathematical functions and asks to find the sum and difference of two functions. It requires evaluating expressions involving f(x) and g(x) and performing algebraic operations.

Answer

- \( f + g = 9x^2 + 5x - 2 \) - \( f - g = x^2 - 9x + 8 \)

Steps to Solve

Identify the Functions The functions given are:

( f(x) = 5x^2 - 2x + 3 )
( g(x) = 4x^2 + 7x - 5 )

Finding ( f + g ) To find ( f + g ), we add the two functions together: [ f(x) + g(x) = (5x^2 - 2x + 3) + (4x^2 + 7x - 5) ] Combine like terms: [ = (5x^2 + 4x^2) + (-2x + 7x) + (3 - 5) ] [ = 9x^2 + 5x - 2 ]
Finding ( f - g ) To find ( f - g ), we subtract ( g(x) ) from ( f(x) ): [ f(x) - g(x) = (5x^2 - 2x + 3) - (4x^2 + 7x - 5) ] Distributing the negative sign: [ = 5x^2 - 2x + 3 - 4x^2 - 7x + 5 ] Combine like terms: [ = (5x^2 - 4x^2) + (-2x - 7x) + (3 + 5) ] [ = x^2 - 9x + 8 ]

The results are:

( f + g = 9x^2 + 5x - 2 )
( f - g = x^2 - 9x + 8 )

More Information

These results illustrate how to perform addition and subtraction of polynomial functions. The structures of the functions allow us to combine and simplify effectively using like terms.

Tips

Forgetting to combine like terms correctly. Always keep track of each term's coefficient.
Misapplying the operations (adding instead of subtracting). Ensure the operations in the expressions are correctly remembered.

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