HAEF IB - MATH HL
TEST 2
FUNCTIONS
by Christos Nikolaidis
Name:______
Date:______
Questions
- [Maximum mark: 7]
Some values of the self inverse function f are given below.
(a)Find
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(ii) ......
(iii) ......
(b) Given that g is an even function, with and , find
(i) .
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(ii) find a solution of the equation
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- [Maximum mark: 5]
Let P be a random point on the graph of f (x) = .
Show that the distance between P and A is equal to the shortest distance between the point P and the line
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- [Maximum mark: 6]
Consider the functions f (x) = x2 – k and g(x) = kx–x2 .
Find the possible values of k in each of the following cases.
(a)Given that the graphs of the functions do not intersect.
(b)Given that the graph of g(x) is the reflection of the graph of f (x) in x-axis
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- [Maximum mark: 10]
The function f is defined by
, for
(a)Write down the equations of the vertical asymptotes of.
(b)Find the equation of the horizontal asymptote and justify your answer.
(c)Show that the integer 1 does not lie in the range of .
(d)Find an expression for .
(e)Given that , solve the equation
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- [Maximum mark: 13]
Consider the function f (x) =
(a)Complete the following table
Function / Horizontal Asymptote / Vertical Asymptotes / y-intercept / max or min/ min at
(-3.74,4.45)
(b)Sketch the graph of f (x) by indicating any asymptotes and intersections with the two axes.
(c)Write down the range of f.
(d)The point A(1,9/7)lies on the graph of . Find the coordinates of theimage point A΄ under the transformation y = 7f (2x–5)
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- [Maximum mark: 5]
The diagram shows the graphs of the functions and .
Sketch the graph of .
Indicate clearly where the y-intercept, the x-intercepts and any asymptotes occur.
- [Maximum mark: 9]
Find in the form y = ax2 + bx + cthe equations of the following quadratic functionsgiven that the corresponding graph
(a)passes through the vertex (2,7) and the y-intercept y= 3.
(b)passes through the x-intercepts x = 2 and x = 3 and the y-intercept y = 6
(c)passes through the origin (0,0), and the points (1,3), (–2,6).
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- [Maximum mark: 10]
Let , .
(a)Find the maximum value of , given that exists
(b)Write down the range of f.
(c)Find the inverse function in the appropriate domain.
(d)Describe a sequence of transformations that maps the graph of ,
to the graph of
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- [Maximum mark: 9]
The graph of the function is shown below
(a)On the same diagram above, sketch the graph of .
(b)On the diagram below, sketch the graph of .
Some values of the function g are given in the following table.
(c)Find the value of
(d)Find a solution of the equation
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- [Maximum mark: 6]
Part of the graph of the function is shown below.
Sketch the graph of the reciprocal of . Indicate any possible asymptotes and intersections with the axes.
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