Introduction to Functions - Algebra 2

Define . As an exponential function, this has a graph that has no vertical asymptote, as is defined for all real values of . In terms of :

,

The graph of is a transformation of that of - a horizontal shift ( ), a vertical stretch ( ), and a vertical shift ( ) of the graph of ; none of these transformations changes the status of the function as one whose graph has no vertical asymptote.

Define in terms of ,

It can be restated as the following:

The graph of has as its horizontal asymptote the line of the equation . The graph of is a transformation of that of —a right shift of 2 units , a vertical stretch , and an upward shift of 5 units . The right shift and the vertical stretch do not affect the position of the horizontal asymptote, but the upward shift moves the asymptote to the line of the equation . This is the correct response.

All inputs are valid. There is nothing you can put in for x that won't work.

is a rational function in simplest form whose denominator has a polynomial with degree greater than that of the polynomial in its numerator (2 and 1, respectively). The graph of such a function has as its horizontal asymptote the line of the equation .

All inputs are valid. There is nothing you can put in for x that won't work.

You cannot take the square root of a negative number.

The function includes all possible y-values (outputs). There is nothing you can put in for y that won't work.

Squaring an input cannot produce a negative output.

The function includes all possible y-values (outputs). There is nothing you can put in for y that won't work.

A number taken to a power must be positive.

The square root of any number cannot be negative.

The absolute value of a number cannot be negative.

To create the inverse, switch x and y making the solution x=3y+3.

y must be isolated to finish the problem.

The domain is all the possible values of . To determine the domain, we need to determine what values don't work for this equation. The only value that is not allowed for this equation is 5, since that would make the denominator have a value of , and you can not divide by . Therefore, the domain of this equation is:

and

The domain of any quadratic function is always all real numbers.

The range of this function is anything greater than or equal to 5.

These written in the correct notation is:

Soft brackets are needed for infinity and a hard square bracket for 5 because it is included in the solution.

Because the domain is giving us a wide range of values, we can easily eliminate the fractional function as it only isolates a single value. We can eliminate as it means I am restricted to as my domain but I am looking for domain values greater than . This leaves us with the radical functions.

We have to remember the smallest possible value inside the radical is zero. Anything less means we will be dealing with imaginary numbers.

, This means the domain is which doesn't match our domain so this is wrong.

, . This means the domain is which doesn't match our domain since we want to EXCLUDE so this is wrong.

Since this is fractional expression with a radical in the denominator, we need to remember the bottom can't be zero and just set that denominator to equal . Square both sides to get . This actually means is not acceptable but any values greater than that is good. This is the correct answer.

All inputs are valid. There is nothing you can put in for x that won't work.

All inputs are valid. There is nothing you can put in for x that won't work.

To find the domain, find all areas of the number line where the fraction is defined.

because the denominator of a fraction must be nonzero.

Factor by finding two numbers that sum to -2 and multiply to 1. These numbers are -1 and -1.