# How do I sketch a graph of a function and find its domain range and the least or greatest value of f(x) and value of x ?

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(1) Sketch a graph:

First determine the type of function. There are a few basic functions that you should be familiar with such as polynomials (including lines), rational functions, exponential functions, logarithmic functions (including logistic functions), power functions, and the basic trigonometric functions.

Once you have determined the type of function, you should have an idea of the basic shape of the graph. (e.g. a quadratic function has a parabola for a graph, etc...) Then create a table of values to find some points on the graph and connect them with a curve. You could also use the concept of transformations to help. (So if you know how to graph `y=x^3` then to graph `y=(x-3)^3-2` you would translate the first graph 3 units to the right and 2 units down; that is every point on the original graph (x,y) is sent to (x+3,y-2).)

(2) Finding the domain is relatively simple once you identify the graph type. The domain is all possible inputs. For polynomials the domain is all real numbers unless you are given some restriction. The same is true for exponentials, power functions, logistic functions, and the sine and cosine.

Some functions come with domain restrictions. The tangent function has repeated discontinuities. The logarithmic functions generally have a vertical asymptote and even root functions (such as the square root function) have domain restrictions in the reals.

But generally to determine the domain you assume all real numbers and then check that you are not: (1) dividing by zero (especially for rational functions), (2) are not taking an even root of a negative number or (3) taking a logarithm of a negative number. If none of these occur, and you are not given any other restrictions then the domain is all real numbers. (A domain restriction looks something like y=x+3 for x>2 where the domain is x>2.)

(3) Finding the range can be a bit more of a challenge. The range is all possible outputs. Again knowing the type of function is critical. The graph can help. For polynomials of odd degree the range is all real numbers as it is with most logarithmic functions and odd root functions. Even degree polynomials have either a maximum or minimum -- in the case of a maximum the range is bounded above (e.g. `y<=2` ). The range of logistics functions is bounded above and below. The range of even rhe range of rational functions can be difficult.

(4) Finding maximums and minimums -- typically you need calculus. There are some functions that you can determine max and min without calculus. For example for `y=x^2+3` the graph is a parabola opening up and thus has a minimum at the vertex or a minimum at (0,3).

If you have calculus, take the first derivative of the function and set it equal to zero. Solving this equation gives you critical points (including any points where the derivative fails to exist.) Then apply the first derivative test to determine if the critical point(s) is/are maximum, minimum, or neither.

Example: Given `y=(x-2)/(x+3)` :

The function is a rational function. In this case it is linear over linear so the graph is a hyperbola. There is a horizontal asymptote at y=1 and a vertical asymptote at x=-3. The domain is `x!=-3` and the range is `y!=1` .(There are various ways to write these: we could write the domain as `-oo,-3)uu(-3,oo)` or ```` -{-3}, etc...)

There are no maximums or minimums, The graph: