Functions - Equations

HAEF IB - MATH HL

TEST 3

FUNCTIONS - EQUATIONS

by Christos Nikolaidis

Name:______

Date:______

Questions

1.[Maximum mark: 7]

The tables below show some values of the functionsf, g and h-1

(a)Write down the values of f -1(3), h(2) [2 marks]

(b)Calculate (fog)(2) [2 marks]

(c)Find a solution of the equation (hof)(x) = 4 [3 marks]

......

2.[Maximum mark: 5]

Find all linear functions of the form , , such that the graph of

f (x) coincides with the graph of . [5 marks]

......

3.[Maximum mark: 6]

There is a point P on the parabola and a point Q on the line , so that the point M(2, is the midpoint of the line segment [PQ]. Find the coordinates of P and Q. [6 marks]

......

4.[Maximum mark: 7]

Consider the function f (x) = kx2 – 2kx+2. Find the values of k

(a)if the equation f (x) = 0 has exactly two solutions. [3 marks]

(b)if the graphs of f (x) and –have exactly one intersection point. [2 marks]

(c)if kx2 +2 > 2kx for any value of x. [2 marks]

......

5.[Maximum mark: 6]

Consider the following polynomial of degree 3,

(a)Find the value of k, given that is a repeated root of [2marks]

(b)Find the values of k, given that has exactly one real root.[4 marks]

......

6.[Maximum mark: 5]

Find the real values of x for which

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7.[Maximum mark: 8]

The polynomial is a factor of

(a)Find the values of and [4 marks]

(b)Find the remainder when is divided by [2 marks]

(c)Solve the equation .[2 marks]

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8.[Maximum mark: 6]

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.

[6 marks]

9.[Maximum mark: 12]

Let = .

(a)Sketch the graphs ofand on the same axes by indicating any possible intercepts, roots, minimum/maximum values and asymptotes.

[6 marks]

(b)Hence complete the following table

Function / / / / /
Domain
Range

[5 marks]

(c)The functionhas no inverse. What is the minimum possible value of a, such that the function f is invertible when defined in the interval (a,+∞)?

[1 mark]

......

10.[Maximum mark: 14]

Let g(x) = and h(x) = .

(a)Solve the equation (goh)(x) = [3 marks]

(b)Solve the equation (hog)(x) = [3 marks]

(c)Find [2 marks]

(d)Find the function f (x) given that gof = h [3 marks]

(e)Find the function f (x) given that fog = h [3 marks]

......

11.[Maximum mark: 14]

Consider the function f (x) =

(a)Sketch the graphs of f (x) and by indicating any asymptotes and intersections with x- and y-axes.

[5 marks]

(b)Complete the following table

Function / / /
Domain
Range

[3 marks]

(c)Find the corresponding position to the horizontal asymptote of ,

under the following transformations:

Transformation / 2f (x) / f (x)+2 / f (x–7) / – f (x)
Horizontal asymptote

[4 marks]

(d)The point A(3,0.5) lies on the graph of . Find the corresponding position to the point A under the transformation

y = 2f (3x)+5 [2 marks]

......

12.[Maximum mark: 10]

Consider the polynomial , where .

(a)Show that for all values of c. [1 mark]

(b)Express as a product of three linear factors. [2 marks]

(c)If the graph passes through the point A(1, 0), determine the value of c and hence sketch the graph of , clearly indicating the x-intercepts. [3 marks]

(d)For any value of c, with, solve the inequality

[4 marks]

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