12 multiple-choice questions, progressively harder.
If f(x)=x2f(x) = x^2f(x)=x2, what is f(x+1)f(x + 1)f(x+1)?
Solution
Correct answer: D
Substitute the whole expression x+1x + 1x+1 for xxx, then expand the square.
f(x+1)=(x+1)2=x2+2x+1f(x + 1) = (x + 1)^2 = x^2 + 2x + 1f(x+1)=(x+1)2=x2+2x+1
The common slip x2+1x^2 + 1x2+1 drops the middle term 2x2x2x.
What is the domain of f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2?
Correct answer: B
The inside of a real square root must be zero or positive.
x−2≥0 ⟹ x≥2x - 2 \ge 0 \;\Longrightarrow\; x \ge 2x−2≥0⟹x≥2
So the domain is all numbers greater than or equal to 222.
If f(x)=x2f(x) = x^2f(x)=x2, how many inputs xxx produce the output f(x)=25f(x) = 25f(x)=25?
Correct answer: C
Solve x2=25x^2 = 25x2=25 for every input that works.
x2=25 ⟹ x=5 or x=−5x^2 = 25 \;\Longrightarrow\; x = 5 \text{ or } x = -5x2=25⟹x=5 or x=−5
Two different inputs give the same output, which a function is allowed to do.
Which equation does NOT define yyy as a function of xxx?
Correct answer: A
Test whether each input xxx gives a single yyy. Only x=y2x = y^2x=y2 fails.
x=4 ⟹ y=2 or y=−2x = 4 \;\Longrightarrow\; y = 2 \text{ or } y = -2x=4⟹y=2 or y=−2
One input with two outputs is not a function. The constant y=5y = 5y=5 is still a function; every input gives the one output 555.
If f(x)=x2−9x−3f(x) = \dfrac{x^2 - 9}{x - 3}f(x)=x−3x2−9, what is f(3)f(3)f(3)?
Check the domain before simplifying. At x=3x = 3x=3 the denominator is zero, so 333 is not in the domain.
f(3)=32−93−3=00 is undefinedf(3) = \frac{3^2 - 9}{3 - 3} = \frac{0}{0} \;\text{is undefined}f(3)=3−332−9=00is undefined
The expression equals x+3x + 3x+3 only for x≠3x \neq 3x=3, so the tempting value 666 is not f(3)f(3)f(3).
If f(x)=x+2f(x) = x + 2f(x)=x+2, what is f(5)−f(1)5−1\dfrac{f(5) - f(1)}{5 - 1}5−1f(5)−f(1)?
Evaluate the two outputs, then divide by the change in input. Here f(5)=7f(5) = 7f(5)=7 and f(1)=3f(1) = 3f(1)=3.
f(5)−f(1)5−1=7−34=1\frac{f(5) - f(1)}{5 - 1} = \frac{7 - 3}{4} = 15−1f(5)−f(1)=47−3=1
If f(x)=3x+bf(x) = 3x + bf(x)=3x+b and f(2)=11f(2) = 11f(2)=11, what is bbb?
Substitute the input 222 and set the result equal to 111111.
3(2)+b=11 ⟹ 6+b=11 ⟹ b=53(2) + b = 11 \;\Longrightarrow\; 6 + b = 11 \;\Longrightarrow\; b = 53(2)+b=11⟹6+b=11⟹b=5
If f(x)=ax+1f(x) = ax + 1f(x)=ax+1 and f(3)=10f(3) = 10f(3)=10, what is aaa?
Substitute the input 333 and solve for aaa.
3a+1=10 ⟹ 3a=9 ⟹ a=33a + 1 = 10 \;\Longrightarrow\; 3a = 9 \;\Longrightarrow\; a = 33a+1=10⟹3a=9⟹a=3
What is the range of f(x)=x2+1f(x) = x^2 + 1f(x)=x2+1?
Start from what x2x^2x2 can do and shift by the added 111.
x2≥0 ⟹ x2+1≥1x^2 \ge 0 \;\Longrightarrow\; x^2 + 1 \ge 1x2≥0⟹x2+1≥1
The smallest output is 111 (at x=0x = 0x=0), so the range is all numbers greater than or equal to 111.
If f(x)=2x+1f(x) = 2x + 1f(x)=2x+1, what is f(x)−f(x−1)f(x) - f(x - 1)f(x)−f(x−1)?
Evaluate both, using f(x−1)=2(x−1)+1=2x−1f(x - 1) = 2(x - 1) + 1 = 2x - 1f(x−1)=2(x−1)+1=2x−1.
f(x)−f(x−1)=(2x+1)−(2x−1)=2f(x) - f(x - 1) = (2x + 1) - (2x - 1) = 2f(x)−f(x−1)=(2x+1)−(2x−1)=2
If g(x)=x2g(x) = x^2g(x)=x2, what is g(a)−g(−a)g(a) - g(-a)g(a)−g(−a)?
Squaring turns opposite inputs into the same output, since g(−a)=(−a)2=a2g(-a) = (-a)^2 = a^2g(−a)=(−a)2=a2.
g(a)−g(−a)=a2−a2=0g(a) - g(-a) = a^2 - a^2 = 0g(a)−g(−a)=a2−a2=0
A relation is the set of pairs {(1,2),(2,4),(3,6),(1,2)}\{(1,2),(2,4),(3,6),(1,2)\}{(1,2),(2,4),(3,6),(1,2)}. Is it a function?
The repeated pair (1,2)(1,2)(1,2) is the same pair, so it adds no new output. Input 111 is still tied to just one output.
1↦2 (the repeat is identical)1 \mapsto 2 \;\text{(the repeat is identical)}1↦2(the repeat is identical)
Every input has exactly one output, so it is a function.
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