Question 1

Some linear functions are power functions.

Accepted Answer

Linear functions of the form f(x) = mx + b, where m and b are constants, can be considered power functions when m is nonzero. Specifically, when m is nonzero, the term mx can be seen as mx^1, which fits the definition of a power function with the power of x being 1.

Question 2

Power regression always applies to data that are not exponential.

Accepted Answer

Power regression can be applied to various types of data, including those that exhibit an exponential relationship when one variable is logarithmically transformed, making the relationship linear in a log-log plot.

Question 3

Power regression does not apply to data that should be fit by a decreasing function.

Accepted Answer

Power regression can be used to fit both increasing and decreasing functions, depending on the sign and value of the power parameter in the model.

Question 4

The homogeneity property applies to power functions but not to exponential functions.

Accepted Answer

The homogeneity property, which states that $f(kx) = k^n f(x)$ for some constant $n$, applies to power functions because scaling the input by a constant $k$ scales the output by $k^n$, where $n$ is the power. However, this property does not apply to exponential functions, where the form is $a^x$, because scaling the input by $k$ does not simply scale the output by a constant factor related to $k$; instead, it affects the exponent in a more complex way.

Deck 23: Modeling Data with Power Functions