12 multiple-choice questions, progressively harder.
If f(x)=2xf(x) = 2xf(x)=2x and g(x)=x+3g(x) = x + 3g(x)=x+3, what is f(g(x))−g(f(x))f(g(x)) - g(f(x))f(g(x))−g(f(x))?
Solution
Correct answer: D
Find each composition first. Here f(g(x))=2(x+3)=2x+6f(g(x)) = 2(x + 3) = 2x + 6f(g(x))=2(x+3)=2x+6 and g(f(x))=2x+3g(f(x)) = 2x + 3g(f(x))=2x+3.
f(g(x))−g(f(x))=(2x+6)−(2x+3)=3f(g(x)) - g(f(x)) = (2x + 6) - (2x + 3) = 3f(g(x))−g(f(x))=(2x+6)−(2x+3)=3
A store first halves an item's price, then takes 101010 dollars off the result. If ppp is the original price, which function gives the final price?
Correct answer: C
Halving is h(p)=p2h(p) = \dfrac{p}{2}h(p)=2p and taking 101010 off is d(x)=x−10d(x) = x - 10d(x)=x−10. Halving happens first, so ddd is the outer function.
d(h(p))=p2−10d(h(p)) = \frac{p}{2} - 10d(h(p))=2p−10
Doing the steps in the other order, p−102\dfrac{p - 10}{2}2p−10, is a different (and cheaper) policy.
If f(x)=x2f(x) = x^2f(x)=x2, what is f(f(x))f(f(x))f(f(x))?
Substitute x2x^2x2 into fff and square it.
f(f(x))=(x2)2=x4f(f(x)) = (x^2)^2 = x^4f(f(x))=(x2)2=x4
Squaring a square multiplies the exponents.
A square has side sss, and its area is A(s)=s2A(s) = s^2A(s)=s2. If the side grows with time as s(t)=2ts(t) = 2ts(t)=2t, what is the area as a function of ttt?
Correct answer: A
Substitute the side rule s(t)=2ts(t) = 2ts(t)=2t into the area rule and square it.
A(s(t))=(2t)2=4t2A(s(t)) = (2t)^2 = 4t^2A(s(t))=(2t)2=4t2
Squaring 2t2t2t squares both the 222 and the ttt.
If f(x)=x−2f(x) = x - 2f(x)=x−2 and g(x)=x2g(x) = x^2g(x)=x2, what is g(f(x))g(f(x))g(f(x))?
The inner function is fff, so square the whole rule x−2x - 2x−2.
g(f(x))=(x−2)2=x2−4x+4g(f(x)) = (x - 2)^2 = x^2 - 4x + 4g(f(x))=(x−2)2=x2−4x+4
The middle term −4x-4x−4x is part of the square.
If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x−4g(x) = x - 4g(x)=x−4, for which input does f(g(x))=0f(g(x)) = 0f(g(x))=0?
Correct answer: B
Build the composition, then set it equal to 000. Here f(g(x))=(x−4)2f(g(x)) = (x - 4)^2f(g(x))=(x−4)2.
(x−4)2=0 ⟹ x−4=0 ⟹ x=4(x - 4)^2 = 0 \;\Longrightarrow\; x - 4 = 0 \;\Longrightarrow\; x = 4(x−4)2=0⟹x−4=0⟹x=4
If f(x)=3x+1f(x) = 3x + 1f(x)=3x+1 and g(x)=x−1g(x) = x - 1g(x)=x−1, what is f(g(x))f(g(x))f(g(x))?
Substitute g(x)=x−1g(x) = x - 1g(x)=x−1 into fff, distribute, and combine constants.
f(g(x))=3(x−1)+1=3x−3+1=3x−2f(g(x)) = 3(x - 1) + 1 = 3x - 3 + 1 = 3x - 2f(g(x))=3(x−1)+1=3x−3+1=3x−2
If f(x)=3xf(x) = 3xf(x)=3x and f(g(x))=6x+9f(g(x)) = 6x + 9f(g(x))=6x+9, what is g(x)g(x)g(x)?
Since fff triples its input, f(g(x))=3⋅g(x)f(g(x)) = 3 \cdot g(x)f(g(x))=3⋅g(x). Set that equal to 6x+96x + 96x+9 and divide by 333.
3⋅g(x)=6x+9 ⟹ g(x)=2x+33 \cdot g(x) = 6x + 9 \;\Longrightarrow\; g(x) = 2x + 33⋅g(x)=6x+9⟹g(x)=2x+3
If f(x)=x2f(x) = x^2f(x)=x2 and g(x)=x−4g(x) = x - 4g(x)=x−4, what is g(f(3))g(f(3))g(f(3))?
The inner function is fff, so square 333 first. Here f(3)=9f(3) = 9f(3)=9.
g(9)=9−4=5g(9) = 9 - 4 = 5g(9)=9−4=5
The other order gives f(g(3))=f(−1)=1f(g(3)) = f(-1) = 1f(g(3))=f(−1)=1, a different result.
If f(x)=x2f(x) = \dfrac{x}{2}f(x)=2x and g(x)=x+5g(x) = x + 5g(x)=x+5, what is f(g(x))f(g(x))f(g(x))?
Substitute g(x)=x+5g(x) = x + 5g(x)=x+5 into fff, which halves its input.
f(g(x))=x+52f(g(x)) = \frac{x + 5}{2}f(g(x))=2x+5
The whole quantity x+5x + 5x+5 is halved, not just the xxx.
If f(x)=x2−4f(x) = x^2 - 4f(x)=x2−4 and g(x)=x+2g(x) = x + 2g(x)=x+2, what is f(g(x))f(g(x))f(g(x)) in simplest form?
Substitute g(x)=x+2g(x) = x + 2g(x)=x+2 into fff, expand the square, then simplify.
f(g(x))=(x+2)2−4=x2+4x+4−4=x2+4xf(g(x)) = (x + 2)^2 - 4 = x^2 + 4x + 4 - 4 = x^2 + 4xf(g(x))=(x+2)2−4=x2+4x+4−4=x2+4x
If f(x)=2xf(x) = 2xf(x)=2x, g(x)=x−3g(x) = x - 3g(x)=x−3, and h(x)=f(g(x))h(x) = f(g(x))h(x)=f(g(x)), what is h(5)h(5)h(5)?
Run ggg on 555 first, then apply fff. Here g(5)=5−3=2g(5) = 5 - 3 = 2g(5)=5−3=2.
h(5)=f(2)=2(2)=4h(5) = f(2) = 2(2) = 4h(5)=f(2)=2(2)=4
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