# Let f(x) = frac{40x^2 - 2x + 1}{x^3 - 3x^2 + 2x} . Find the domain of f(x) . Find the...

## Question:

Let {eq}f(x) = \frac{40x^2 - 2x + 1}{x^3 - 3x^2 + 2x} {/eq}.

Find the domain of {eq}f(x) {/eq}.

Find the horizontal and vertical asymptotes.

## Concavity analysis:

The analysis of the concavity and the points of inflection is a good tool and gives us great information for drawing curves

A mathematical function is a relationship that is established between two sets, through which each element of the first set is assigned a single element of the second set or none. The domain is where this function is defined.

The asymptote is an imaginary line to which the function tends and which we represent with a broken line These can be of three types.

- Horizontal asymptotes

- Vertical asymptotes

- Oblique asymptotes

Some considerations to understand the exercise.

1. Definition of vertical asymptote

Let a function {eq}f: R \rightarrow R {/eq} and be a point {eq}a \in R {/eq} so that there exists {eq}r > 0 \,\,\,\,\, \textrm {with} \,\,\,\,\, (a-r, a + r) \subset Domain f {/eq}. We say that f has a vertical asymptote on the line {eq}x = a {/eq}

if {eq}\lim_{x \rightarrow a+} f(x) = \pm \infty \,\,\,\,\, \textrm {or} \,\,\,\,\, \lim_{x \rightarrow a-} f(x) = \pm \infty {/eq}

2. Definition of horizontal asymptote

Let a function {eq}f: \subseteq R \rightarrow R {/eq}. We say that f has a horizontal asymptote on the line {eq}y = b {/eq}

if {eq}\lim_{x \rightarrow \infty} f(x) = b \,\,\,\,\, \textrm {or} \,\,\,\,\, \lim_{x \rightarrow - \infty} f(x) = b {/eq}

3. Domain of a function

It is the set formed by the elements that have an image. The values that define the function and form the set

of departure.

Step 1. Find the domain of the function

{eq}f(x)= \frac{40x^2 -2x+1}{x^3 -3x^2+2x} \,\,\,\,\, \textrm {A fractional function is not defined at the...

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