Functions and Mathematical Models Free Online FlashCards

What condition defines a function?

A function assigns exactly one output to each allowed input.

What is the domain of f(x)=1/(x−2)?

The domain is x≠2, or (-∞,2)∪(2,∞), because a denominator cannot equal zero.

How does y=a f(b(x−h))+k transform f?

In y=a f(b(x−h))+k, h shifts horizontally, k shifts vertically, |a| changes vertical scale, and |b| changes horizontal scale inversely.

How is (f∘g)(x) evaluated?

Composition is (f∘g)(x)=f(g(x)); g acts first, followed by f.

When does a function have an inverse function?

A function must be one-to-one, meaning distinct inputs produce distinct outputs, to have an inverse function.

What is the period of a sinusoidal function?

For y=A sin(B(x−C))+D or y=A cos(B(x−C))+D, the period is 2π/|B|.

Where does tan x have vertical asymptotes?

Tangent is undefined where cos x=0, at x=π/2+kπ for k∈ℤ.

What does the sign of k mean in P(t)=P₀e^{kt}?

In P(t)=P₀e^{kt}, k>0 indicates continuous growth and k<0 indicates continuous decay.

What is the relationship between ln x and e^x?

The natural logarithm is the inverse of e^x: ln(e^x)=x and e^{ln x}=x for x>0.

When is s(t)=t³−6t²+9t momentarily at rest?

The particle is at rest at t=1 and t=3 because v(t)=3(t−1)(t−3) equals zero there.

What are the essential steps in mathematical modeling?

A complete model interpretation defines variables and units, identifies a relationship, determines the domain, analyzes behavior, checks reasonableness, and communicates the result in context.

What are the domain and range of a function?

The domain is the set of permitted inputs; the range is the set of resulting outputs.