2 Functions as Mathematical Models Free Online FlashCards

What makes a relation a function?

A function assigns exactly one output to each permitted input. Its domain contains the inputs, and its range contains the resulting outputs.

What are the domain and range?

The domain is the set of allowable inputs, while the range is the set of resulting outputs.

What does $C(m)=3+2m$ model?

In the taxi model, the starting fee is represented by 3 and the per-mile charge by 2, so the model is $C(m)=3+2m$.

What do $m$ and $b$ mean in $f(x)=mx+b$?

For $f(x)=mx+b$, $m$ is the constant rate of change, and $b=f(0)$ is the vertical intercept.

When is a linear model appropriate?

A linear model is appropriate when equal changes in the input produce approximately equal changes in the output.

What defines a polynomial function?

A polynomial has nonnegative integer exponents, such as $f(x)=x^2-4x+3$. Its natural domain is all real numbers.

How do you identify a quadratic vertex?

In $f(x)=a(x-h)^2+k$, the vertex is $(h,k)$. The sign of $a$ determines whether the parabola opens upward or downward.

What do $a$ and $b$ mean in $f(x)=ab^x$?

In $f(x)=ab^x$, $a$ is the initial value and $b$ is the multiplication factor for each increase of one input unit.

How does $b$ determine exponential growth or decay?

An exponential model represents growth when $b>1$ and decay when $0<b<1$.

How are logarithms related to exponentials?

The logarithmic equation $y=\log_b(x)$ is equivalent to $b^y=x$. A logarithm is the inverse of the corresponding exponential function.

How do you evaluate a piecewise function?

A piecewise function uses different formulas on different parts of its domain. Select the condition containing the input, then apply only that formula.

What are the domain and range of $f(x)=\sqrt{x-4}$?

For $f(x)=\sqrt{x-4}$, require $x-4\ge 0$. Thus the domain is $[4,\infty)$ and the range is $[0,\infty)$.