12 multiple-choice questions, progressively harder.
Evaluate f(−2)f(-2)f(−2) for f(x)=2xf(x) = 2^xf(x)=2x.
Solution
Correct answer: B
A negative exponent gives the reciprocal of the positive power.
2−2=122=142^{-2} = \frac{1}{2^2} = \frac{1}{4}2−2=221=41
The output is a positive fraction, never a negative number.
Which function represents decay?
Correct answer: C
Decay needs a base strictly between 000 and 111.
0<23<10 < \tfrac{2}{3} < 10<32<1
The bases 333, 32\tfrac{3}{2}23, and 222 are all greater than 111, so they grow.
Evaluate f(12)f\left(\tfrac{1}{2}\right)f(21) for f(x)=9xf(x) = 9^xf(x)=9x.
Correct answer: D
A one-half power is a square root.
91/2=9=39^{1/2} = \sqrt{9} = 391/2=9=3
Between equally spaced inputs, an exponential function has a constant what?
Each equal step multiplies the output by the base, so consecutive outputs share a constant ratio.
f(x+1)f(x)=b\frac{f(x+1)}{f(x)} = bf(x)f(x+1)=b
A constant difference is the signature of a linear function instead.
Evaluate 4−14^{-1}4−1.
Correct answer: A
A negative exponent gives a reciprocal.
4−1=144^{-1} = \frac{1}{4}4−1=41
For f(x)=100⋅(0.5)xf(x) = 100 \cdot (0.5)^xf(x)=100⋅(0.5)x, find f(2)f(2)f(2).
Square the base, then multiply by 100100100.
f(2)=100⋅(0.5)2=100⋅0.25=25f(2) = 100 \cdot (0.5)^2 = 100 \cdot 0.25 = 25f(2)=100⋅(0.5)2=100⋅0.25=25
At which input do 2x2^x2x and x2x^2x2 give the same value?
Compare the two functions at each input.
2x∣x=2=4=x2∣x=22^x\big|_{x=2} = 4 = x^2\big|_{x=2}2xx=2=4=x2x=2
At x=2x = 2x=2 both equal 444. The other inputs differ: 222 vs 111, then 888 vs 999, then 323232 vs 252525.
For f(x)=bxf(x) = b^xf(x)=bx, which base makes the function increase the fastest for large xxx?
Among growth bases, a larger base multiplies by more at each step, so it climbs faster.
5>2>1.1>15 > 2 > 1.1 > 15>2>1.1>1
The base 12\tfrac{1}{2}21 is less than 111, so it decays rather than grows.
For f(x)=a⋅bxf(x) = a \cdot b^xf(x)=a⋅bx with a=3a = 3a=3 and b=4b = 4b=4, find f(1)f(1)f(1).
Substitute and use 41=44^1 = 441=4.
f(1)=3⋅41=3⋅4=12f(1) = 3 \cdot 4^1 = 3 \cdot 4 = 12f(1)=3⋅41=3⋅4=12
Which statement about f(x)=2xf(x) = 2^xf(x)=2x is TRUE?
A positive base raised to any real power stays positive, so the graph lives above the axis.
2x>0 for every x2^x > 0 \;\text{for every } x2x>0for every x
Also f(0)=1f(0) = 1f(0)=1, and the graph is a curve, not a line.
The values 3,12,48,1923, 12, 48, 1923,12,48,192 form an exponential pattern. What is the constant ratio?
Divide each term by the one before it.
123=4,4812=4,19248=4\frac{12}{3} = 4, \quad \frac{48}{12} = 4, \quad \frac{192}{48} = 4312=4,1248=4,48192=4
The ratio is 444.
Evaluate (13)−1\left(\tfrac{1}{3}\right)^{-1}(31)−1.
A negative exponent flips the fraction.
(13)−1=1 13 =3\left(\tfrac{1}{3}\right)^{-1} = \frac{1}{\,\tfrac{1}{3}\,} = 3(31)−1=311=3
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