Question

50 POINTS!!!

Write two linear functions, f(x) and g(x). For example, f(x) = 3x – 7 and g(x) = –2x + 5. Then see whether f(x) – (–g(x)) is equivalent to f(x)+ g(x). Hint: To find –g(x), just change the signs of all the terms in g(x). Discuss whether you think your results would apply to every function.

This can be any equation that would work for this, there is no specific answer.

Answer 1

f(x) = x

g(x) = -2x+1

f(x) -(-g(x)) = -x+1

f(x) +g(x) = -x+1

f(x)-(-g(x)) = (f+g)(x) is true for all functions f and g, linear or not

Explanation:

We can define a couple of linear functions as ...

f(x) = x

g(x) = -2x+1

Then the reflected function -g(x) is ...

-g(x) = -(-2x +1) = 2x -1

And the difference from f(x) is ...

f(x) -(-g(x)) = x -(2x -1) = -x +1 . . . . f(x) -(-g(x))

We want to compare that to the sum of the functions:

f(x) +g(x) = x +(-2x +1) = -x +1 . . . . f(x) +g(x)

The two versions of the function expression have the same value.

These results are a property of addition, so do not depend on the nature of f(x) or g(x). They will hold for every function.

Answer 2

9514 1404 393

Answer:

• f(x) = x
• g(x) = -2x+1
• f(x) -(-g(x)) = -x+1
• f(x) +g(x) = -x+1
• f(x)-(-g(x)) = (f+g)(x) is true for all functions f and g, linear or not

Explanation:

We can define a couple of linear functions as ...

f(x) = x

g(x) = -2x+1

Then the reflected function -g(x) is ...

-g(x) = -(-2x +1) = 2x -1

And the difference from f(x) is ...

f(x) -(-g(x)) = x -(2x -1) = -x +1 . . . . f(x) -(-g(x))

We want to compare that to the sum of the functions:

f(x) +g(x) = x +(-2x +1) = -x +1 . . . . f(x) +g(x)

The two versions of the function expression have the same value.

These results are a property of addition, so do not depend on the nature of f(x) or g(x). They will hold for every function.