One over a function

In this lesson $\displaystyle f(x)$ is a polynomial of degree at most 2, and $\displaystyle {g(x)}={\frac{1}{f(x)}}$.
The value of $\displaystyle g$ at each $\displaystyle x$ is 1 divided by the value of $\displaystyle f$ at the same $\displaystyle x$:

• $\displaystyle {f(x)}={2}$ – then $\displaystyle {g(x)}={\frac{1}{2}}$;
• $\displaystyle {f(x)}={-4}$ – then $\displaystyle {g(x)}={-\frac{1}{4}}$;
• $\displaystyle {f(x)}={0.01}$ – then $\displaystyle {g(x)}={100}$: a small value of $\displaystyle f$ gives a large value of $\displaystyle g$.

So the shape of $\displaystyle g$’s graph can be told from the properties of $\displaystyle f$, without a derivative.

• The domain: every $\displaystyle x$ except the zeros of $\displaystyle f$ – there is no dividing by 0.
• No points on the $\displaystyle x$-axis: a fraction whose numerator is 1 is never 0.
• Positive and negative: $\displaystyle g$ is positive exactly where $\displaystyle f$ is positive, and negative where $\displaystyle f$ is negative.
• Rising and falling: on every interval where $\displaystyle f$ is not 0, where $\displaystyle f$ rises $\displaystyle g$ falls, and where $\displaystyle f$ falls $\displaystyle g$ rises – a larger denominator gives a smaller fraction.
• The extremum point: at the vertex of $\displaystyle f$, if it is not on the $\displaystyle x$-axis, $\displaystyle g$ has an extremum point of the opposite kind: a minimum of $\displaystyle f$ with value $\displaystyle m$ is a maximum of $\displaystyle g$ with value $\displaystyle \frac{1}{m}$, and the other way round.
• Common points: where $\displaystyle {f(x)}={1}$ or $\displaystyle {f(x)}={-1}$, $\displaystyle g(x)$ equals the same number, and the graphs meet.

An asymptote is a line that the graph comes closer and closer to as it goes along it. $\displaystyle g$ has two kinds of asymptote, parallel to the axes:

• A vertical asymptote: near a zero $\displaystyle {x}={a}$ of $\displaystyle f$, the values of $\displaystyle f$ are close to 0, so the values of $\displaystyle g$ are very far from 0 – very large, positive or negative. The graph comes closer to the line $\displaystyle {x}={a}$ and never touches it.
• A horizontal asymptote: when $\displaystyle x$ becomes very large ($\displaystyle x\to\infty$) or very small ($\displaystyle x\to-\infty$), a polynomial of degree 1 or 2 moves far from 0, so $\displaystyle g$ comes close to 0. The line $\displaystyle {y}={0}$ – the $\displaystyle x$-axis – is a horizontal asymptote.

$\displaystyle {f(x)}={x^2-4x+3}={(x-1)(x-3)}$, and $\displaystyle {g(x)}={\frac{1}{x^2-4x+3}}$.
In the example below we find the properties of $\displaystyle g$ from those of $\displaystyle f$:

• The zeros of $\displaystyle f$: $\displaystyle {x}={1}$, $\displaystyle {x}={3}$. The domain of $\displaystyle g$: $\displaystyle x\neq1$, $\displaystyle x\neq3$.
• The asymptotes of $\displaystyle g$: $\displaystyle {x}={1}$, $\displaystyle {x}={3}$ (vertical), $\displaystyle {y}={0}$ (horizontal).
• $\displaystyle f>0$ when $\displaystyle x<1$ or $\displaystyle x>3$, so $\displaystyle g>0$ there too; when $\displaystyle 1<x<3$, $\displaystyle g<0$.
• The vertex of $\displaystyle f$: $\displaystyle (2,-1)$, a minimum. $\displaystyle g$ has a maximum there, $\displaystyle {g(2)}={\frac{1}{-1}}={-1}$ – the same point.
• $\displaystyle f$ falls when $\displaystyle x<2$, so $\displaystyle g$ rises when $\displaystyle x<1$ and when $\displaystyle 1<x<2$; $\displaystyle g$ falls when $\displaystyle 2<x<3$ and when $\displaystyle x>3$.

The dashed lines are the vertical asymptotes. The graph of $\displaystyle g$ (the second colour) never crosses the $\displaystyle x$-axis, and on each side of a vertical asymptote it runs off upwards or downwards.

$\displaystyle {f(x)}={-x^2+7x-12}$, and $\displaystyle {g(x)}={\frac{1}{f(x)}}$.

Solve the equation $\displaystyle {f(x)}={0}$, and write the domain of $\displaystyle g$.

Write the equations of the three asymptotes of $\displaystyle g$ on the board.

The vertex of $\displaystyle f$ is halfway between its zeros. Mark the extremum point of $\displaystyle g$.

On which intervals is $\displaystyle g$ positive, and on which negative? On which intervals does it rise, and on which does it fall?

On the board – the graph of the function $\displaystyle f(x)$ on the interval $\displaystyle -0.5\leq x\leq2.5$.

Write the intervals where $\displaystyle f(x)$ and $\displaystyle \frac{1}{f(x)}$ rise and fall.

On the graph of $\displaystyle f$, mark the points where $\displaystyle f(x)$ and $\displaystyle \frac{1}{f(x)}$ have the same value.

What are the extremum points of $\displaystyle \frac{1}{f(x)}$ on this interval?

$\displaystyle {f(x)}={x^2+1}$.
Write $\displaystyle {g(x)}={\frac{1}{f(x)}}$ on the board, and mark its extremum point.

$\displaystyle {f(x)}={x-2}$ – a polynomial of degree 1, and $\displaystyle {g(x)}={\frac{1}{x-2}}$.
Write the equations of the asymptotes of $\displaystyle g$ on the board.

$\displaystyle f$ rises everywhere. On which intervals does $\displaystyle g$ rise or fall? Does it have an extremum point?