Inverse functions:
1i) Let f(x)=ln(x+3) sketch the mirror line of y=x on the axes. Sketch f(x) on the axes.
ii) Indicate the domain and range of f(x).
iii) Is f(x) a one-to-one function? Explain this algebraically and graphically by using appropriate tests.
iv) Find the inverse function of f(x).Show all steps and sketch and label the inverse function on the previousgraph of question 1i.
v) Indicate the domain and range of the inverse function.Compare the domain and range of f(x) and of the inverse f(x);what statement can be made about them.
vi) Let g be the inverse function in part iv. Show that g o f(x)=x and g o f(y)=y (show all your steps.)
2) Use trigonometric identities to establish the following identity:
Show that cos(x+pi/2)=-sin(x)
3a) Explain what each of the following mathematical statements mean in regard to the graph of f(x).With your explanation, include a DIAGRAM for each part. Also Indicate whether the statement refers to a VERTICAL ASYMPTOTE, HORIZONTAL ASYMPTOTE, or NEITHER. State the equation of the asymptote where applicable.
3a=
i)f(1)=0
ii)lim (X-->0) f(x)=-infinity
iii)lim(x-->2+) f(x)= - inifinity
iv)lim (x-->2-) f(x)=infinity
v)lim(+ or - infinity) f(x)=0
3b) Use your explanation from part 3a and SKETCH the graph of this function.USE dotted lines TO INDICATE ANY ASYMPTOTES.Label all parts of the graph including roots and equations of asymptotes.
3c) Find a formula for a rational function found in part 3a. BRIEFLY explain how you determined this function.
3d) Let f(x)=x^4+x-3. Explain why their must be at least onepoint c in [1,2] satisfying f(c)=0.Indicate which theorem you use in your explanation.
4) Evaluate lim(x-->infinity) ln(x^50)/x^100.Please reference any theorems and examples that you used to answer this question.
4b) A hiker leaves his cabin at 7:00am and takes his usual path to the top of the mountain, arriving at 7:00pm.The following morning he starts at 7:00am at the top and takes the same path back ,arriving at his cabin at 7:00pm.USE THE INTERMEDIATE VALUE THEOREM to explain that their is point on the path that the hiker will cross exactly at the same time of day at both days.
5) Write one mathematical statement that means THE DISTANCE FROM 3 TO X IS LESS THAN 1/4 AND X=3 (THE EQUAL SIGN HAS A LINE THROUGH IT)
5b)=60 degrees=___?_____Radians.
5c)4pi/3 radians=___?_____Degrees? Justify you answer.
5d)What is the RANGE of f(x)=e^x?. DRAW a Sketch of f(x). LABEL theVALUE of they-intercept.
5e)For xless than or equal to x less than or equal to pi/2, solve sin(x)=1/2? JUSTIFY YOUR ANSWER.
5f)Solve arctan(-1)=? Justify your answer.
Answer each of these questions either=
TRUE OR FALSE
6i)cos(3pi/4)=1/root2
ii)Every function is either an even or odd function?
iii)The function cos(x) is invertible on [-pi/2,pi/2]?
iv)The function e^x is the inverse of the function
ln(x)?
v)A sequence {a n} can have more than one limit?
vi) The sequence {1,-1,1,-1} hastwo limits?
vii) f(x)=x^2-1/x-1 and g(x)=x+1 are the same functions?
viii)If lim(x-->a+) f(x) and lim (x-->a-) f(x) exists,then so does lim(x-->a) f(x)?
xi)lim(x-->infinity) ln(x)/x^p=0?
x)lim(x-->0) sin(theta)/theta