Answer:
a) f(x) = -5/2x +50
b) f(x) = 50·0.90^x
c) f(x) = 8/3x -30
d) f(x) = 10·1.15^x
Step-by-step explanation:
Apparently, you're studying linear and exponential functions. The general form of a linear function in "slope-intercept" form is ...
f(x) = mx + b
where m is the slope (rate of change of output with respect to input), and b is the "output-intercept", the function value for an input of zero.
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The general form of an exponential function is ...
f(x) = a·(1 +r)^x
where "a" is the "output-intercept", the function value for an input of zero, and r is the fractional change for a unit change in input, x. Here, you're given r as a percentage. It will be positive when output increases with input, and it will be negative when output decreases with input.
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Using the above function prototypes in the given scenarios, we have ...
a. A linear function with b=50, m=-10/4 = -5/2, so the function is ...
f(x) = -5/2x +50
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b. An exponential function with a=50, r=-0.10, so the function is ...
f(x) = 50·(1 -0.10)^x
f(x) = 50·0.90^x
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c. A linear function with b=-30 and m = 16/6 = 8/3, so the function is ...
f(x) = 8/3x -30
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d. An exponential function with a=10 and r=+0.15, so the function is ...
f(x) = 10·1.15^x
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<em>Comment on functions with percentage change</em>
You need to be a little bit careful with percentage change in a function. There are some scenarios where the rate of change in a linear function is described using percentages. For example, one might say that a car depreciates linearly, and has lost 40% of its value in the first 3 years. Then the rate of change is <em>-13.33% of its original value</em> each year.
Percentages are always in relation to some base value. If that base value is changing with each percentage change, the function is exponential. If that base value remains constant, then the function is linear.
