# 1.5 Exponential and logarithmic functions  (Page 10/17)

 Page 10 / 17

h

g

Domain: $x>5,$ range: all real numbers

$h\circ f$

$g\circ f$

Domain: $x>2$ and $x<-4,$ range: all real numbers

Find the degree, y -intercept, and zeros for the following polynomial functions.

$f\left(x\right)=2{x}^{2}+9x-5$

$f\left(x\right)={x}^{3}+2{x}^{2}-2x$

Degree of 3, $y$ -intercept: 0, zeros: 0, $\sqrt{3}-1,-1-\sqrt{3}$

Simplify the following trigonometric expressions.

$\frac{{\text{tan}}^{2}x}{{\text{sec}}^{2}x}+{\text{cos}}^{2}x$

$\text{cos}\left(2x\right)={\text{sin}}^{2}x$

$\text{cos}\left(2x\right)$ or $\frac{1}{2}\left(\text{cos}\left(2x\right)+1\right)$

Solve the following trigonometric equations on the interval $\theta =\left[-2\pi ,2\pi \right]$ exactly.

$6{\text{cos}}^{2}x-3=0$

${\text{sec}}^{2}x-2\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}x+1=0$

$0,±2\pi$

Solve the following logarithmic equations.

${5}^{x}=16$

${\text{log}}_{2}\left(x+4\right)=3$

4

Are the following functions one-to-one over their domain of existence? Does the function have an inverse? If so, find the inverse ${f}^{-1}\left(x\right)$ of the function. Justify your answer.

$f\left(x\right)={x}^{2}+2x+1$

$f\left(x\right)=\frac{1}{x}$

One-to-one; yes, the function has an inverse; inverse: ${f}^{-1}\left(x\right)=\frac{1}{y}$

For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain.

$f\left(x\right)=\sqrt{9-x}$

$f\left(x\right)={x}^{2}+3x+4$

$x\ge -\frac{3}{2},{f}^{-1}\left(x\right)=-\frac{3}{2}+\frac{1}{2}\sqrt{4y-7}$

A car is racing along a circular track with diameter of 1 mi. A trainer standing in the center of the circle marks his progress every 5 sec. After 5 sec, the trainer has to turn 55° to keep up with the car. How fast is the car traveling?

For the following problems, consider a restaurant owner who wants to sell T-shirts advertising his brand. He recalls that there is a fixed cost and variable cost, although he does not remember the values. He does know that the T-shirt printing company charges $440 for 20 shirts and$1000 for 100 shirts.

a. Find the equation $C=f\left(x\right)$ that describes the total cost as a function of number of shirts and b. determine how many shirts he must sell to break even if he sells the shirts for $10 each. a. $C\left(x\right)=300+7x$ b. 100 shirts a. Find the inverse function $x={f}^{-1}\left(C\right)$ and describe the meaning of this function. b. Determine how many shirts the owner can buy if he has$8000 to spend.

For the following problems, consider the population of Ocean City, New Jersey, which is cyclical by season.

The population can be modeled by $P\left(t\right)=82.5-67.5\phantom{\rule{0.1em}{0ex}}\text{cos}\left[\left(\pi \text{/}6\right)t\right],$ where $t$ is time in months $\left(t=0$ represents January 1) and $P$ is population (in thousands). During a year, in what intervals is the population less than 20,000? During what intervals is the population more than 140,000?

The population is less than 20,000 from December 8 through January 23 and more than 140,000 from May 29 through August 2

In reality, the overall population is most likely increasing or decreasing throughout each year. Let’s reformulate the model as $P\left(t\right)=82.5-67.5\phantom{\rule{0.1em}{0ex}}\text{cos}\left[\left(\pi \text{/}6\right)t\right]+t,$ where $t$ is time in months ( $t=0$ represents January 1) and $P$ is population (in thousands). When is the first time the population reaches 200,000?

For the following problems, consider radioactive dating. A human skeleton is found in an archeological dig. Carbon dating is implemented to determine how old the skeleton is by using the equation $y={e}^{rt},$ where $y$ is the percentage of radiocarbon still present in the material, $t$ is the number of years passed, and $r=-0.0001210$ is the decay rate of radiocarbon.

If the skeleton is expected to be 2000 years old, what percentage of radiocarbon should be present?

78.51%

Find the inverse of the carbon-dating equation. What does it mean? If there is 25% radiocarbon, how old is the skeleton?

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
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
how do i evaluate integral of x^1/2 In x
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
by using integration product formula
Roha
find derivative f(x)=1/x
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
do u mean fundamental theorem ?
Roha
I want simple integral
for MSc chemistry... simple formulas of integration
aparna
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funny
how are you
funny
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aparna
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aparna
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funny
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funny
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funny
I just formulas
aparna
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aparna
value of log ax cot-x cos-x
aparna
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funny
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funny
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aparna
I am M.sc physics now i am studying in m.phil
funny
so what can i do
aparna
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funny
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funny
integration f(X) dx this is basic formula of integration sign is not there you can look integration sign in methematics form... and f(X) my be any function any values
funny
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funny
funny
Hi
RIZWAN
I don't understand the formula
who's formula
funny
which formula?
Roha
what is the advantages of mathematical economics
Mubarak