# 1.2 Basic classes of functions  (Page 7/28)

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Find the domain for each of the following functions: $f\left(x\right)=\left(5-2x\right)\text{/}\left({x}^{2}+2\right)$ and $g\left(x\right)=\sqrt{5x-1}.$

The domain of $f$ is $\text{(−∞, ∞).}$ The domain of $g$ is $\left\{x|x\ge 1\text{/}5\right\}.$

## Transcendental functions

Thus far, we have discussed algebraic functions. Some functions, however, cannot be described by basic algebraic operations. These functions are known as transcendental functions because they are said to “transcend,” or go beyond, algebra. The most common transcendental functions are trigonometric, exponential, and logarithmic functions. A trigonometric function relates the ratios of two sides of a right triangle. They are $\mathrm{sin}x,\mathrm{cos}x,\mathrm{tan}x,\mathrm{cot}x,\mathrm{sec}x,\text{and}\phantom{\rule{0.2em}{0ex}}\mathrm{csc}x.$ (We discuss trigonometric functions later in the chapter.) An exponential function is a function of the form $f\left(x\right)={b}^{x},$ where the base $b>0,b\ne 1.$ A logarithmic function    is a function of the form $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ for some constant $b>0,b\ne 1,$ where ${\mathrm{log}}_{b}\left(x\right)=y$ if and only if ${b}^{y}=x.$ (We also discuss exponential and logarithmic functions later in the chapter.)

## Classifying algebraic and transcendental functions

Classify each of the following functions, a. through c., as algebraic or transcendental.

1. $f\left(x\right)=\frac{\sqrt{{x}^{3}+1}}{4x+2}$
2. $f\left(x\right)={2}^{{x}^{2}}$
3. $f\left(x\right)=\text{sin}\left(2x\right)$
1. Since this function involves basic algebraic operations only, it is an algebraic function.
2. This function cannot be written as a formula that involves only basic algebraic operations, so it is transcendental. (Note that algebraic functions can only have powers that are rational numbers.)
3. As in part b., this function cannot be written using a formula involving basic algebraic operations only; therefore, this function is transcendental.

Is $f\left(x\right)=x\text{/}2$ an algebraic or a transcendental function?

Algebraic

## Piecewise-defined functions

Sometimes a function is defined by different formulas on different parts of its domain. A function with this property is known as a piecewise-defined function    . The absolute value function is an example of a piecewise-defined function because the formula changes with the sign of $x\text{:}$

$f\left(x\right)=\left\{\begin{array}{c}\text{−}x,x<0\\ x,x\ge 0\end{array}.$

Other piecewise-defined functions may be represented by completely different formulas, depending on the part of the domain in which a point falls. To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for $x and $x>a,$ we need to pay special attention to what happens at $x=a$ when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at $x=a.$ We examine this in the next example.

## Graphing a piecewise-defined function

Sketch a graph of the following piecewise-defined function:

$f\left(x\right)=\left\{\begin{array}{l}x+3,\phantom{\rule{3em}{0ex}}x<1\\ {\left(x-2\right)}^{2},\phantom{\rule{1.5em}{0ex}}x\ge 1\end{array}.$

Graph the linear function $y=x+3$ on the interval $\left(\text{−∞},1\right)$ and graph the quadratic function $y={\left(x-2\right)}^{2}$ on the interval $\left[1,\infty \right).$ Since the value of the function at $x=1$ is given by the formula $f\left(x\right)={\left(x-2\right)}^{2},$ we see that $f\left(1\right)=1.$ To indicate this on the graph, we draw a closed circle at the point $\left(1,1\right).$ The value of the function is given by $f\left(x\right)=x+2$ for all $x<1,$ but not at $x=1.$ To indicate this on the graph, we draw an open circle at $\left(1,4\right).$

#### Questions & Answers

find the domain and range of f(x)= 4x-7/x²-6x+8
find the range of f(x)=(x+1)(x+4)
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
d/dx{1/y - lny + X^3.Y^5}
How to identify domain and range
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
sorry
Dr
Dr
:(
Shun
was up
Dr
hello
is it chatting app?.. I do not see any calculus here. lol
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
show that lim f(x) + lim g(x)=m+l
list the basic elementary differentials
Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie