04/14 4. Functions and Their Graphs Free Online FlashCards

What condition makes a relation a function?

A function is a relation in which every input is paired with exactly one output. Different inputs may share an output.

How can a graph be tested for being a function?

The vertical line test: a graph represents a function of $x$ if every vertical line intersects it at no more than one point.

What are the domain and range of a relation?

The domain is the set of inputs, and the range is the set of outputs. In $(x,y)$, the $x$-values form the domain and the $y$-values form the range.

Find the range of $R=\{(-2,5),(0,1),(3,5),(4,8)\}$.

For $R=\{(-2,5),(0,1),(3,5),(4,8)\}$, the range is $\{1,5,8\}$. The repeated output $5$ appears only once.

What is the domain of a graph open at $-3$ and closed at $4$?

The domain is $(-3,4]$: $-3$ is excluded because its endpoint is open, and $4$ is included because its endpoint is closed.

Why is $x=2$ excluded from $f(x)=\frac{1}{x-2}$?

For $f(x)=\frac{1}{x-2}$, the denominator cannot be zero, so $x\ne 2$. The domain is $(-\infty,2)\cup(2,\infty)$.

What is the domain of $g(x)=\sqrt{x+5}$?

For $g(x)=\sqrt{x+5}$, require $x+5\ge 0$. Thus the domain is $[-5,\infty)$.

What does $f(x)$ mean?

Function notation names the output produced by a function for a given input. It is read “$f$ of $x$,” not “$f$ times $x$.”

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

For $f(x)=3x^2-2x+1$, $f(2)=3(2)^2-2(2)+1=9$.

Find $f(a+1)$ for $f(x)=x^2+4x$.

If $f(x)=x^2+4x$, then $f(a+1)=(a+1)^2+4(a+1)=a^2+6a+5$.

How does adding a constant outside $f(x)$ transform its graph?

The function $g(x)=f(x)+k$ shifts the graph up $k$ units when $k>0$; $g(x)=f(x)-k$ shifts it down $k$ units.

How does $f(x-h)$ transform a graph?

The expression $f(x-h)$ shifts a graph right $h$ units, while $f(x+h)$ shifts it left $h$ units. The inside sign acts oppositely.