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C3 logs/functions questions
Could somebody please help me with the following questions:
7) The points P and Q lie on the curve with equation y=e^1/2x. The x-coordinates of P and Q are ln 4 and ln 16 respectively.
a) Find an equation for the line PQ.
b) Show that this line passes through the origin O.
c) Calculate the length, to 3 significant figures, of the line segment PQ.
6) The functions f and g are given by f:x >>> 3x-1 (xER)
g:x >>> e^x/2 (xER)
a) Find the value of gf(4), giving your answer to 2 decimal places.
b) Express the inverse function f^-1(x) in the form f^-1:x >>> …
c) Using the same axes, sketch the graphs of the functions f and gf. Write on your sketch the value of each function at x=0.
d) Find the values of x for which f^-1 (x) = 5/f(x).
5) The function f is defined by:
f:x >>> ln (5x-2) (xER, x>2/5)
a) Find an expression for f^-1(x).
b) Write down the domain of f^-1(x)
c) Solve, giving your answers to 3 decimal places, ln (5x-2) =2.
4) The price of a computer system can be modelled by the formula: P=100+850e^-t/2 where P is the price of the system in £s and t is the age if the computer in years after being purchased.
a) Calculate the new price of the system.
b) Calculate its price after 3 years.
c) When will it be worth less than £200?
d) Find its price as t>>>∞
e) Sketch the graph showing P against t.
Comment on the appropriateness of this model.
7) The points P and Q lie on the curve with equation y=e^1/2x. The x-coordinates of P and Q are ln 4 and ln 16 respectively.
a) Find an equation for the line PQ.
b) Show that this line passes through the origin O.
c) Calculate the length, to 3 significant figures, of the line segment PQ.
6) The functions f and g are given by f:x >>> 3x-1 (xER)
g:x >>> e^x/2 (xER)
a) Find the value of gf(4), giving your answer to 2 decimal places.
b) Express the inverse function f^-1(x) in the form f^-1:x >>> …
c) Using the same axes, sketch the graphs of the functions f and gf. Write on your sketch the value of each function at x=0.
d) Find the values of x for which f^-1 (x) = 5/f(x).
5) The function f is defined by:
f:x >>> ln (5x-2) (xER, x>2/5)
a) Find an expression for f^-1(x).
b) Write down the domain of f^-1(x)
c) Solve, giving your answers to 3 decimal places, ln (5x-2) =2.
4) The price of a computer system can be modelled by the formula: P=100+850e^-t/2 where P is the price of the system in £s and t is the age if the computer in years after being purchased.
a) Calculate the new price of the system.
b) Calculate its price after 3 years.
c) When will it be worth less than £200?
d) Find its price as t>>>∞
e) Sketch the graph showing P against t.
Comment on the appropriateness of this model.
Reply 1
turtle2
5) The function f is defined by:
f:x >>> ln (5x-2) (xER, x>2/5)
a) Find an expression for f^-1(x).
b) Write down the domain of f^-1(x)
c) Solve, giving your answers to 3 decimal places, ln (5x-2) =2.
f:x >>> ln (5x-2) (xER, x>2/5)
a) Find an expression for f^-1(x).
b) Write down the domain of f^-1(x)
c) Solve, giving your answers to 3 decimal places, ln (5x-2) =2.
Here is my solution
Reply 2
turtle2
4) The price of a computer system can be modelled by the formula: P=100+850e^-t/2 where P is the price of the system in £s and t is the age if the computer in years after being purchased.
a) Calculate the new price of the system.
b) Calculate its price after 3 years.
c) When will it be worth less than £200?
d) Find its price as t>>>?
e) Sketch the graph showing P against t.
Comment on the appropriateness of this model.
a) Calculate the new price of the system.
b) Calculate its price after 3 years.
c) When will it be worth less than £200?
d) Find its price as t>>>?
e) Sketch the graph showing P against t.
Comment on the appropriateness of this model.
Here is my solution to question 4. The rest I will leave for others.
Reply 3
turtle2
OP
7
How would u solve:
d) ln x + ln (x-3) = 0
e) e^x + e^-x = 2
f) ln 2 + ln x = 4
the inverse function of 3+ln(4-x)?
d) ln x + ln (x-3) = 0
e) e^x + e^-x = 2
f) ln 2 + ln x = 4
the inverse function of 3+ln(4-x)?
Reply 4
turtle2
How would u solve:
d) ln x + ln (x-3) = 0
e) e^x + e^-x = 2
f) ln 2 + ln x = 4
the inverse function of 3+ln(4-x)?
d) ln x + ln (x-3) = 0
e) e^x + e^-x = 2
f) ln 2 + ln x = 4
the inverse function of 3+ln(4-x)?
Here is solution for (d)
Reply 5
turtle2
How would u solve:
d) ln x + ln (x-3) = 0
e) e^x + e^-x = 2
f) ln 2 + ln x = 4
the inverse function of 3+ln(4-x)?
d) ln x + ln (x-3) = 0
e) e^x + e^-x = 2
f) ln 2 + ln x = 4
the inverse function of 3+ln(4-x)?
Solutions (e) and (f)
Reply 6
turtle2
OP
7
thanks! what about the inverse function one?
Reply 7
turtle2
OP
7
for d), the answer is 3 + √13 / 2 ?
Reply 8
turtle2
for d), the answer is 3 + ?13 / 2 ?
I think it is 3/2 + ?13/2 .... there might be a misprint
Reply 9
turtle2
OP
7
mm thanks. what about the inverse function of 3+ln(4-x)?
Reply 10
turtle2
thanks! what about the inverse function one?
Here is my solution but I know it is wrong. Can someone tell me my mistake?
I know it is wrong because the graph is not symmetrical about the line y=x
Edit: Solution is correct, graph had the wrong scale to show symmetry
See pic 9
Reply 11
turtle2
OP
7
steve2005
Here is my solution but I know it is wrong. Can someone tell me my mistake?
I know it is wrong because the graph is not symmetrical about the line y=x
I know it is wrong because the graph is not symmetrical about the line y=x
In the book it says the answer is 4-e^x-3, so that's right...but yea, the graph is a bit strange...
Reply 12
turtle2
OP
7
5) The function f is defined by:
F:x >>> ln(x-2), xER, x>2.
(There’s a sketch of the curve with eqn y=f(x). The curve crosses the x-axis at the point P(p,0). The curve has an asymptote, shown by a broken line in the diagram, whose eqn is x=q.
a) Write down the value of p and the value of q.
b) Find the function f^-1 and state its domain.
c) Sketch the curve with equation y=f^-1 (x) and its asymptote.
Write on your sketch the coordinates of any point where the curve crosses the coordinate axes and the equation of the asymptote.
6) A formula used to calculate the power gain of an amplifier has the form:
G=h ln (p2 / p1)
Given that G=16, h=4.3 and p1=8,
a) Calculate, to the nearest whole number, the value of p2.
Given that the values of G and p1 are exact but that the value of h has been given to one decimal place,
b) Find he range of possible values of p2.
F:x >>> ln(x-2), xER, x>2.
(There’s a sketch of the curve with eqn y=f(x). The curve crosses the x-axis at the point P(p,0). The curve has an asymptote, shown by a broken line in the diagram, whose eqn is x=q.
a) Write down the value of p and the value of q.
b) Find the function f^-1 and state its domain.
c) Sketch the curve with equation y=f^-1 (x) and its asymptote.
Write on your sketch the coordinates of any point where the curve crosses the coordinate axes and the equation of the asymptote.
6) A formula used to calculate the power gain of an amplifier has the form:
G=h ln (p2 / p1)
Given that G=16, h=4.3 and p1=8,
a) Calculate, to the nearest whole number, the value of p2.
Given that the values of G and p1 are exact but that the value of h has been given to one decimal place,
b) Find he range of possible values of p2.
Reply 13
turtle2
OP
7
7) The function is defined by f:x >>>e^x + k, xER and k is a positive constant.
a) State the range of f.
b) Find f(ln k), simplifying your answer.
c) Find f^-1, the inverse function of f, in the form f^-1:x >>>…, stating its domain.
d) On the same axes, sketch the curves with equations y=f(x) and y=f^-1(X), giving your coordinates of all points where the graphs cut the axes.
8) The functions f and g are defined over the set of real numbers by:
f:x >>>3x-5
g:x >>>e^-2x
a) State the range of g.
b) Sketch the graphs of the inverse functions f^-1 and g^-1 and write on your sketches the coordinates of any points at which a graph meets the coordinates axes.
c) State, giving a reason, the number of roots of the equation: f^-1(x)=g^-1(x).
d) Evaluate fg(-1/3), giving your answer to decimal places.
9) The function f is defined by:
f:x >>>ln (5x-2), x>2/5
a) Find an expression for f^-1(x).
b) Write down the domain of f^-1.
c) Solve, giving your answer to 3 decimal places, ln (5x-2)=2.
a) State the range of f.
b) Find f(ln k), simplifying your answer.
c) Find f^-1, the inverse function of f, in the form f^-1:x >>>…, stating its domain.
d) On the same axes, sketch the curves with equations y=f(x) and y=f^-1(X), giving your coordinates of all points where the graphs cut the axes.
8) The functions f and g are defined over the set of real numbers by:
f:x >>>3x-5
g:x >>>e^-2x
a) State the range of g.
b) Sketch the graphs of the inverse functions f^-1 and g^-1 and write on your sketches the coordinates of any points at which a graph meets the coordinates axes.
c) State, giving a reason, the number of roots of the equation: f^-1(x)=g^-1(x).
d) Evaluate fg(-1/3), giving your answer to decimal places.
9) The function f is defined by:
f:x >>>ln (5x-2), x>2/5
a) Find an expression for f^-1(x).
b) Write down the domain of f^-1.
c) Solve, giving your answer to 3 decimal places, ln (5x-2)=2.
Reply 14
You expect us to do your homework for u! No point in copying out the questions and posting them!Try to understand them and finding the solutions yourself!
Reply 15
15
Bye You expect us to do your homework for u! No point in copying out the questions and posting them!
Try to understand them and finding the solutions yourself!
You expect us to do your homework for u! No point in copying out the questions and posting them!Try to understand them and finding the solutions yourself!
I totally agree - you learn more by struggling with a question for half an hour (even with small hints) (and reading back over the theory) than you do by getting a solution and thinking 'oh yes I get it now'.
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