01 Functions and Their Graphs Free Online FlashCards

What is a function?

A function assigns exactly one output to each allowed input. The allowed inputs form the domain, and the resulting outputs form the range.

What does the vertical line test determine?

A graph represents a function of $x$ if every vertical line intersects it at most once.

Evaluate $f(3)$ for $f(x)=x^2-4x+1$.

Substitute $3$ for every occurrence of $x$: $f(3)=3^2-4(3)+1=-2$.

Find the domain of $g(x)=\sqrt{x+2}$.

The domain is $[-2,\infty)$, because a real even root requires $x+2\ge 0$.

What are the intercepts of $f(x)=x^2-4$?

For $f(x)=x^2-4$, the $y$-intercept is $(0,-4)$, and the $x$-intercepts are $(-2,0)$ and $(2,0)$.

What does the horizontal line test determine?

A function is one-to-one if every horizontal line intersects its graph at most once.

How does $g(x)=(x-3)^2$ transform $y=x^2$?

The graph shifts right $3$ units because the input is replaced by $x-3$.

Describe the transformations in $g(x)=-2(x-1)^2+4$.

Starting from $y=x^2$, shift right $1$, stretch vertically by $2$, reflect across the $x$-axis, and shift up $4$.

In what order is $(f\circ g)(x)$ evaluated?

Apply $g$ first and then $f$: $(f\circ g)(x)=f(g(x))$.

Find $(f\circ g)(x)$ when $f(x)=2x+1$ and $g(x)=x^2$.

For $f(x)=2x+1$ and $g(x)=x^2$, $(f\circ g)(x)=2x^2+1$.

What conditions define the domain of $f\circ g$?

An input must be in the domain of $g$, and its output $g(x)$ must be in the domain of $f$.

When does a function have an inverse function?

A function must be one-to-one on its domain to have an inverse function.