Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# f(x)=ax^2-5b For the function f defined above, a and b are constants and f(x+2)=5. What is the value of f(-x-2)? a) -5 b) 0 c) 5 d) 10?

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• The answer is as follows:

5 (c)

• f(x) = ax^2 - 5b. f(x+2) = 5, ie., a(x+2)^2 -5b = 5;

f(-x-2)= f(-(x+2)) = a(-(x+2))^2 -5b = a(x+2)^2 -5b = f(x+2) = 5. Answer = c).

•  f(x) = ax^2 - 5b

For the function f defined above, a and b are constants

and f(x + 2) = 5.

The value of f(-x - 2):

a) -5

• There's really no reason to expect (a) to be the answer.  The only use of x in f(x) is to square it, and there's no simple way to tweak x to make the sign of f(x) flip, considering that -5b is going to be added.

What I'd guess you're supposed to see is that the f(x) is a polynomial with only even powers of x, which means that f(-x) = f(x) for all x.  Such a function sometimes called "even", which is short for "symmetrical with respect to the x=0 line".

Then, I'll bet you're supposed to recognize that -x-2 = -(x + 2).  Put those together and:

f(-x - 2) = f(-[x+2])  . . . simplifying the argument to f

= f(x + 2)   . . . because f is "even"

= 5            . . . because f(x+2) = 5 was given

Write these out until you are comfortable with the pattern.  Soon, you'll be able to answer a question like this with essentially no calculation at all, just by recognizing symmetry.

If a polynomial f(x) has only odd powers of x instead, then you'd have f(-x) = -f(x).  Such a function is called "odd", short for "antisymmetrical with respect to the x=0 line".  In that case, if f(x + 2) = 5 were true, then f(-x - 2) would be -5.  Again, almost no calculation is needed...just a few sign flips.