+24 1. The table of values represents a reciprocal function f(x).
How much greater is the average rate of change over the interval [−6, −4][−6, −4] than the interval [−3, −1][−3, −1] ?
x f(x)
−6 −0.0046
−5 −0.0079
−4 −0.0154
−3 −0.0357
−2 −0.1111
−1 −0.5
2. Consider two functions: g(x)=20(1.5)x and the function f(x) shown in the table.
Which statements are true?
Select each correct answer.
x f(x)
−5 −45
−4 −48
−3 −49
−2 −48
−1 −45
0 −40
1 −33
a) g(x) has a greater y-intercept than f(x) does.
b) f(x) increases at a faster rate than g(x) does on the interval (−5, −3) .
c) f(1) is less than g(−1) .
d) f(x) and g(x) are both increasing on the interval (−∞, ∞) .
+9466 1. Here's a graph of the points to get an idea of what the function looks like.
average rate of change = \(\frac{\text{change in }f(x)}{\text{change in }x}\)
average rate of change over the interval [-6, -4] = \( \frac{ f(-6) \,-\, f(-4)}{ (-6) \,-\,( -4) }\)
\( \frac{ f(-6) \,-\, f(-4)}{ (-6) \,-\,( -4) }\,=\,\frac{ (-0.0046) \,-\, (-0.0154) }{ (-6) \,-\, (-4) }\,=\,\frac{ 0.0108 }{-2 }\,=\,- 0.0054\)
average rate of change over the interval [-6, -4] = - 0.0054
Notice that this is just the slope of the line through the points ( -6, f(-6) ) and ( -4, f(-4) ) .
average rate of change over the interval [-3, -1] = \( \frac{ f(-3) \,-\, f(-1)}{ (-3) \,-\,( -1) }\)
\( \frac{ f(-3) \,-\, f(-1)}{ (-3) \,-\,( -1) }\,=\, \frac{ (-0.0357) \,-\, (-0.5)}{ (-3) \,-\,( -1) }\,=\, \frac{0.4643}{ -2 }\,=\,- 0.23215\)
average rate of change over the interval [-3, -1] = - 0.23215
How much greater is - 0.0054 than - 0.23215 ?
(-0.0054) - (-0.23215) = 0.22675
+9466 2. Assuming that g(x) = 20(1.5)x
the y-intercept of g(x) = g(0) = 20(1.5)0 = 20(1) = 20
the y-intercept of f(x) = f(0) = -40
The y-intercept of g(x) is greater than the y-intercept of f(x) , so a) is true.
From -5 to -3 , f(x) goes from -45 to -49 , and (-49) - (-45) = -4
From -5 to -3 , g(x) goes from \(\frac{640}{243}\) to \(\frac{160}{27}\) , and \(\frac{160}{27}\) - \(\frac{640}{243}\) = \(\frac{800}{243}\)
f(x) is decreasing on the interval (-5, -3) because as x gets larger, f(x) gets smaller.
\(\frac{800}{243}\) > -4 , so g(x) is increasing faster than f(x) on the interval (-5, -3) . b) is false.
f(1) = -33
g(-1) = 20(1.5)-1 = \(\frac{20}{1.5}\) = \(\frac{40}{3}\) ≈ 13.3
Is f(1) less than g(-1) ? Is -33 less than 13.3 ? Yes, so c) is true.
We already found that f(x) is decreasing on the interval (-5, -3) , so
f(x) can't be increasing on the interval (−∞, ∞) . d) is false.
