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The function f is defined by [m]f(x) = ax^4 4x^2 + ax 3[/m] for al
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03 Mar 2022, 04:00
03 Mar 2022, 04:00
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The function f is defined by \(f(x) = ax^4 – 4x^2 + ax – 3\) for all positive numbers x. What will \(f(b) – f(–b)\) equal?
A) \(0\)
B) \(2ab\)
C) \(–8b^2 – 6\)
D) \(2ab^4 –8b^2 – 6\)
E) \(2ab^4 –8b^2 + 2ab – 6\)
A) \(0\)
B) \(2ab\)
C) \(–8b^2 – 6\)
D) \(2ab^4 –8b^2 – 6\)
E) \(2ab^4 –8b^2 + 2ab – 6\)
Show: ::
When you see f(x) notation, you have a function. The f(x) part—read as f-of-x—tells you what to plug into the function for x. For example, if they ask you to find f(3) for that function, then plug 3 into the equation everywhere you see x and calculate your answer.
In this case, the problem asks you to find something with f(b) and f(–b). The same principle applies. Plug in b for x and then plug in –b for x, like this:
f(b) = ab4 – 4b2 + ab – 3
f(–b) = ab4 – 4b2 – ab – 3
In the second line, the first two terms end up staying the same because you're raising the –b to even powers, so the values turn positive. The only place that the –b makes a difference is in the third term, ab. In this case, you end up subtracting the ab term rather than adding it.
Finally, the problem asks you to subtract: f(b) – f(–b)
ab4 – 4b2 + ab – 3
– (ab4 – 4b2 – ab – 3)
That subtraction sign has to be carried through every term in the second line. It's a good idea to rewrite the whole thing to minimize careless mistakes:
ab4 – 4b2 + ab – 3
– ab4 + 4b2 + ab + 3
0 + 0 + 2ab + 0
The ab4 terms cancel out. The 4b2 terms cancel out. Even the +/– 3 terms cancel out! The only thing that doesn't cancel out is ab + ab = 2ab.
The correct answer is (B).
In this case, the problem asks you to find something with f(b) and f(–b). The same principle applies. Plug in b for x and then plug in –b for x, like this:
f(b) = ab4 – 4b2 + ab – 3
f(–b) = ab4 – 4b2 – ab – 3
In the second line, the first two terms end up staying the same because you're raising the –b to even powers, so the values turn positive. The only place that the –b makes a difference is in the third term, ab. In this case, you end up subtracting the ab term rather than adding it.
Finally, the problem asks you to subtract: f(b) – f(–b)
ab4 – 4b2 + ab – 3
– (ab4 – 4b2 – ab – 3)
That subtraction sign has to be carried through every term in the second line. It's a good idea to rewrite the whole thing to minimize careless mistakes:
ab4 – 4b2 + ab – 3
– ab4 + 4b2 + ab + 3
0 + 0 + 2ab + 0
The ab4 terms cancel out. The 4b2 terms cancel out. Even the +/– 3 terms cancel out! The only thing that doesn't cancel out is ab + ab = 2ab.
The correct answer is (B).
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The function f is defined by [m]f(x) = ax^4 4x^2 + ax 3[/m] for al
[#permalink]
03 Mar 2022, 06:26
03 Mar 2022, 06:26
Carcass wrote:
The function f is defined by \(f(x) = ax^4 – 4x^2 + ax – 3\) for all positive numbers x. What will \(f(b) – f(–b)\) equal?
A) \(0\)
B) \(2ab\)
C) \(–8b^2 – 6\)
D) \(2ab^4 –8b^2 – 6\)
E) \(2ab^4 –8b^2 + 2ab – 6\)
A) \(0\)
B) \(2ab\)
C) \(–8b^2 – 6\)
D) \(2ab^4 –8b^2 – 6\)
E) \(2ab^4 –8b^2 + 2ab – 6\)
\(f(b) – f(–b) =\) \([a(b)^4 – 4(b)^2 + a(b) – 3]\) - \([a(-b)^4 – 4(-b)^2 + a(-b) – 3]\)
\(=\) \([ab^4 – 4b^2 + ab – 3]\) - \([ab^4 – 4b^2 - ab – 3]\)
\(= 2ab\)
Answer: B
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The function f is defined by [m]f(x) = ax^4 4x^2 + ax 3[/m] for al
[#permalink]
07 Aug 2022, 09:52
07 Aug 2022, 09:52
Given that \(f(x) = ax^4 – 4x^2 + ax – 3\) and we need to find the value of f(b) – f(–b)
To find f(b) we need to compare what is inside the bracket in f(b) and f(x)
=> We need to substitute x with b in \(f(x) = ax^4 – 4x^2 + ax – 3\) to get the value of f(b)
=> \(f(b) = ab^4 – 4b^2 + a*b – 3\) = \(ab^4 – 4b^2 + ab – 3\)
Similarly, \(f(-b) = a(-b)^4 – 4(-b)^2 + a*(-b) – 3\) = \(ab^4 – 4b^2 - ab – 3\)
=> f(b) - f(-b) = \(ab^4 – 4b^2 + ab – 3\) - (\(ab^4 – 4b^2 - ab – 3\))
= \(ab^4 - ab^4 – 4b^2 + 4b^2 + ab + ab – 3 + 3\) = 2ab
So, Answer will be B
Hope it helps!
Watch the following video to learn the Basics of Functions and Custom Characters
To find f(b) we need to compare what is inside the bracket in f(b) and f(x)
=> We need to substitute x with b in \(f(x) = ax^4 – 4x^2 + ax – 3\) to get the value of f(b)
=> \(f(b) = ab^4 – 4b^2 + a*b – 3\) = \(ab^4 – 4b^2 + ab – 3\)
Similarly, \(f(-b) = a(-b)^4 – 4(-b)^2 + a*(-b) – 3\) = \(ab^4 – 4b^2 - ab – 3\)
=> f(b) - f(-b) = \(ab^4 – 4b^2 + ab – 3\) - (\(ab^4 – 4b^2 - ab – 3\))
= \(ab^4 - ab^4 – 4b^2 + 4b^2 + ab + ab – 3 + 3\) = 2ab
So, Answer will be B
Hope it helps!
Watch the following video to learn the Basics of Functions and Custom Characters
