1.1 Functions - Introduction to Calculus

Function

Some types of functions: linear, parabolas, ...

The domain of a function $f$ is the set of all valid input values. The range consists of the set of all output values that can be reached using those domain values.

Rational Function

basic form: $\displaystyle y = {\dfrac{1}{x}}$
vertical asymptote at $x = 0$
horizontal asymptote at $y = 0$

Adjust $\displaystyle y=\frac{a}{x-h}+k$ and move the input to follow a point on the curve.

\displaystyle y=\frac{1}{x}

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Domain:

\displaystyle x\ne 0

Move x toward h from both sides. The dashed line marks x = h; the curve approaches y = k far from h.

Root Function

basic form: $y = \sqrt[n]{x}$
$n$ even: undefined over the real numbers when the radicand is negative
$n$ odd: $x \in \mathbb{R}$

Adjust $\displaystyle y=a\sqrt[n]{x-h}+k$ and move the input to follow a point on the curve.

Root index n

\displaystyle y=\sqrt{x}

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Domain:

\displaystyle x\ge 0

Switch between n = 2 and n = 3, then try an input below h. Even roots exclude it; odd roots remain real.

QUESTION

Which inputs belong to the real domain of

\displaystyle y=\sqrt{x}

Higher-degree Polynomial Functions

basic form: $y = x^n$
domain: $x \in \mathbb{R}$
$n$ even: both ends of the function tend to $+\infty$ or both tend to $-\infty$
$n$ odd: one end tends to $+\infty$ while the other tends to $-\infty$

Adjust $\displaystyle y=a(x-h)^n+k$ and move the input to follow a point on the curve.

Degree n

\displaystyle y=x^{2}

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Domain:

\displaystyle x\in\mathbb{R}

Compare n = 2 with n = 3. Change the sign of a and observe how the two ends of the curve change direction.

For $y=x^n$ , take $n$ to be a positive integer. The statement about both ends tending to $-\infty$ applies when an even-degree polynomial has a negative leading coefficient.

Rational Function

basic form: $\displaystyle y = {\dfrac{1}{x}}$

vertical asymptote at $x = 0$

horizontal asymptote at $y = 0$

Adjust $\displaystyle y=\frac{a}{x-h}+k$ and move the input to follow a point on the curve.

\displaystyle y=\frac{1}{x}

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Domain:

\displaystyle x\ne 0

Move x toward h from both sides. The dashed line marks x = h; the curve approaches y = k far from h.

Root Function

basic form: $y = \sqrt[n]{x}$

$n$ even: undefined over the real numbers when the radicand is negative

$n$ odd: $x \in \mathbb{R}$

Adjust $\displaystyle y=a\sqrt[n]{x-h}+k$ and move the input to follow a point on the curve.

Root index n

\displaystyle y=\sqrt{x}

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Domain:

\displaystyle x\ge 0

Switch between n = 2 and n = 3, then try an input below h. Even roots exclude it; odd roots remain real.

QUESTION

Which inputs belong to the real domain of

\displaystyle y=\sqrt{x}

Higher-degree Polynomial Functions

basic form: $y = x^n$

domain: $x \in \mathbb{R}$

$n$ even: both ends of the function tend to $+\infty$ or both tend to $-\infty$

$n$ odd: one end tends to $+\infty$ while the other tends to $-\infty$

Adjust $\displaystyle y=a(x-h)^n+k$ and move the input to follow a point on the curve.

Degree n

\displaystyle y=x^{2}

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Domain:

\displaystyle x\in\mathbb{R}

Compare n = 2 with n = 3. Change the sign of a and observe how the two ends of the curve change direction.

For

y=x^n

, take

n

to be a positive integer. The statement about both ends tending to

-\infty

applies when an even-degree polynomial has a negative leading coefficient.

Functions

Function

Rational Function

Explore a rational function

Root Function

Explore even and odd roots

KNOWLEDGE QUIZ

Higher-degree Polynomial Functions

Explore polynomial powers

Functions

Function

Rational Function

Explore a rational function

Root Function

Explore even and odd roots

KNOWLEDGE QUIZ

Higher-degree Polynomial Functions

Explore polynomial powers