Because we cannot take the square root of a negative number, the domain of
$\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is
$\text{\hspace{0.17em}}\left(-\infty ,3\right].\text{\hspace{0.17em}}$ Now we check the domain of the composite function
$$(f\circ g)(x)=\sqrt{\sqrt{3-x}+2}$$
The domain of this function is
$\text{\hspace{0.17em}}\left(-\infty ,5\right].\text{\hspace{0.17em}}$ To find the domain of
$\text{\hspace{0.17em}}f\circ g,\text{\hspace{0.17em}}$ we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since
$\text{\hspace{0.17em}}\left(-\infty ,3\right]\text{\hspace{0.17em}}$ is a proper subset of the domain of
$\text{\hspace{0.17em}}f\circ g.\text{\hspace{0.17em}}$ This means the domain of
$\text{\hspace{0.17em}}f\circ g\text{\hspace{0.17em}}$ is the same as the domain of
$\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ namely,
$\text{\hspace{0.17em}}\left(-\infty ,3\right].$
Decomposing a composite function into its component functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to
decompose a composite function , so we may choose the decomposition that appears to be most expedient.
Decomposing a function
Write
$\text{\hspace{0.17em}}f(x)=\sqrt{5-{x}^{2}}\text{\hspace{0.17em}}$ as the composition of two functions.
We are looking for two functions,
$\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}h,\text{\hspace{0.17em}}$ so
$\text{\hspace{0.17em}}f(x)=g(h(x)).\text{\hspace{0.17em}}$ To do this, we look for a function inside a function in the formula for
$\text{\hspace{0.17em}}f(x).\text{\hspace{0.17em}}$ As one possibility, we might notice that the expression
$\text{\hspace{0.17em}}5-{x}^{2}\text{\hspace{0.17em}}$ is the inside of the square root. We could then decompose the function as
$$h(x)=5-{x}^{2}\text{and}g(x)=\sqrt{x}$$
We can check our answer by recomposing the functions.
We can perform algebraic operations on functions. See
[link] .
When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
The function produced by combining two functions is a composite function. See
[link] and
[link] .
The order of function composition must be considered when interpreting the meaning of composite functions. See
[link] .
A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
A composite function can be evaluated from a table. See
[link] .
A composite function can be evaluated from a graph. See
[link] .
A composite function can be evaluated from a formula. See
[link] .
The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See
[link] and
[link] .
Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
Functions can often be decomposed in more than one way. See
[link] .
Section exercises
Verbal
How does one find the domain of the quotient of two functions,
$\text{\hspace{0.17em}}\frac{f}{g}?\text{\hspace{0.17em}}$
Find the numbers that make the function in the denominator
$\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ equal to zero, and check for any other domain restrictions on
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ such as an even-indexed root or zeros in the denominator.
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic.
Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation
of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15)
it's standard equation is x^2 + y^2/16 =1
tell my why is it only x^2? why is there no a^2?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.