 # 3. Analysis of a real function

### Analysis of a real function

We will find and study:

• the domain and zeroes
• continuity, limits in the points in which function is not continuous and at the ends of intervals,
• asymptotes,
• differentiability and derivatives,
• intervals of monotonicity and extrema
• second derivative, convexity, inflection points
• the table of the function,
• parity, periodicity,
• sketch of the graph,
• range.

### An example

We will study .

#### the domain and zeroes

The denominator , if . Therefore, . Moreover, iff .

#### Continuity, limits in the points in which function is not continuous and at the ends of intervals,

The function is continuous on and .

Furthermore:      #### Asymptotes

Therefore we have vertical asymptotes i . There are no horizontal asymptotes.

We check oblique asymptotes:  Therefore, is the right oblique asymptote.  Therefore, is also the left oblique asymptote.

#### Differentiability and derivatives

The funtion in differentiable on the whole domain and: #### Intervals of monotonicity and extrema if or . Therefore:

• on we have , so decreases,
• on we have , so increases,
• on we have , so increases,
• on we have , so increases,
• on we have , so increases,
• on we have , so decreases.

Therefore, in the function has its local minimum, and in its local maximum.

#### Second derivative, convexity, inflection points • on we have , so is convex,
• on we have , so is concave,
• on we have , so is convex,
• on we have , so is concave.

Therefore is an inflection point.

#### The table of the function      decreasing, convex   local minimum   increasing, convex vertical asymptote   increasing, concave   inflection point   increasing, convex vertical asymptote   increasing, concave   local maximum   decreasing, wklęsła.

#### parity, periodicity

The function is odd, since . Therefore it is not even, because it is not a constant zero function. Obviously, it is not a periodic function.

#### Sketch of the graph #### Range

Obviously, .