Functions Examination Questions HL

On this page you can find examination questions from the topic of functions

Equation of a Straight Line

Question 1

The line \(l_1\) has equation 2x+5y+6=0

The line \(l_2\) is perpendicular to the line \(l_1\)and passes through the point (2 , -2)

a) Find the equation of \(l_2\)in the form ax+by+d=0, where a, b and d are constants.

b) Find the coordinates where \(l_2\) meets the y axis


Hint

Full Solution

Question 2

The point A has coordinates (a , 3) and the point B has coordinates (7 , b).

The line AB has equation 2x + 3y = 11.

a) Find the values of a and b

The line AC is perpendicular to the line AB.

b) Find the equation of the line AC in the form ax + by + d = 0, where a, b and d are constants

c) Given that C lies on the x axis, find its coordinates.

Hint

Full Solution

 

 Question 3

The line \(l_1 \) passes through the point P(3k , 2k) with gradient = -2.

\(l_1 \) meets the x axis at A and the y axis at B.

a) Find the equation of the line \(l_1 \) and show that A(4k , 0)

b) Find the area of the triangle AOB in terms of k

The line \(l_2\) passes through P and is perpendicular to \(l_1 \)

c) Find the equation of \(l_2\)

\(l_2\)meets the x axis at C

d) Show that the midpoint of PC lies on the line y = x


Hint

Full Solution

 

Question 4

Point A has coordinates (a , 6) and point B has coordinates (5 , b).

The line 8x - 6y + 3 = 0 is the perpendicular bisector of AB.

Find a and b.

Hint

Full Solution

 

Functions - The Basics

Question 1

The function f and g are defined as follows

\(f(x)=3x^2 \ ,\ x\in\mathbb{R}\)

\(g\left(x\right)=\frac{2}{x}-4\ ,x\in\mathbb{R},\ x>0\)

a) Find the range of g

b) Find g-1

c) Solve gf(x) = 2

Hint

Full Solution

 

Question 2

Let f and g be the functionsLet f and g be the functions

\(f(x)=ln(1-2x),x<\frac{1}{2}\)

\(g(x)=\frac{4}{x-2},x\neq2\)

a) Find the exact value of fg(-2)

b) Find f-1(x), stating its domain

c) Show that \(gg(x)=\frac{4x-8}{8-2x}\)

Hint

Full Solution

Question 3

The graph of the function f is plotted below

Use the graph to find values of

a) \(ff(-2)\)

b) \(f^{-1}(0)\)

Let g be the function such that g(x) = 4x + 2

c) Work out fg(1)

d) Draw a sketch of \(y=f^{-1}(x)\)

Hint

Full Solution

 

Question 4

Let \(f(x)=\frac{ax-1}{x-a},\ x\neq\ a\)

a) Show that \(f^{-1}(x)=f(x)\)

b) Hence, find ff(b)

c) The graph of y = g(x) is shown below. Find the values of a if fg(3) = a + 5

Hint

Full Solution

 

Question 5

Let the function f be defined by

\(f(x)=\frac{x-k}{2-x},x\neq2\)

a) Find \(f^{-1}(x)\) in terms of k

b) Find k if \(y=f(x)\) and \(y=f^{-1}(x)\) intersect at (-2 , -2) and (3 , 3)

Hint

Full Solution

 

Quadratics

Question 1

The graph below is the function f(x)=a(x-h)²+k

The graph pass through the point (0,-9) and has vertex (2,3)

a) Write down the value of a, h and k

b) Find f(x) giving your answer in the form f(x) = Ax²+Bx+C

Hint

Full Solution

 

Question 2

Let f(x) = 2x²+12x +11

The function can also be expressed in the form f(x) = a(x-h)² +k

a) Find the equation of the axis of symmetry

b) Write down the value of h

c) Write down the value of k

Hint

Full Solution

 

Question 3

Let f(x) = x² + 2px + (3p+4)

Find the value of p so that f(x)=0 has two equal roots.

Hint

Full Solution

 

Question 4

Let f(x)=2x² + kx + 1 and g(x) = -x – 1.

The graphs of f and g intersect at two distinct points.

Find the possible values of k

Hint

Full Solution

 

Question 5

A quadratic function f can be written in the form f(x) = a(x - h)² + k . The graph of f has vertex (-1 , 4) and has y-intercept at (0 , 5).

a) Find the value of a , h and k.

b) The line y = mx + 1 is a tangent to the curve f. Find the value of m.

Hint

Full Solution

 

Rational Functions

Question 1

Let f(x) = 2x + 1 and \(g(x)=\frac{x}{1-x} \ ,x\neq1\)

a) Show that \(f\circ g(x)=\frac{x+1}{1-x}\)

b) Let \(h(x)=\frac{x+1}{1-x}\) , for x < 1

c) Sketch the graph of h

d) Sketch the graph of \(h^{-1}\)

Hint

Full Solution

Question 2

Let \(f(x)=\frac{3x-2}{x-a},x\neq\ a\)

a) Find the inverse function \(f^{-1}(x)\) in terms of a

b) Find the value of a such that f is a self-inverse function

Hint

Full Solution

 

Question 3

The function f is defined by \(f(x)=\frac{6x+1}{2x-1},x\in\mathbb{R},x\neq\frac{1}{2}\)

a) Write f(x) in the form \(A+\frac{B}{2x-1}\) where A and B are constants

b) Sketch the graph of f(x) stating the equations of any asymptotes and the coordinates of any intercepts with the axes

Hint

Full Solution

 

Question 4

Sketch the graph of \(f\left(x\right)=\frac{x^2+x-1}{x+2}\) giving the equations of any asymptotes and the coordinates of the x and y intercepts as well as any stationary points


Hint

Full Solution

 

Question 5

Find the value(s) of a such that \(f(x)=\frac{x+1}{ax^2+3x+2}\) has only one vertical asymptote.

Hint

Full Solution

 

Factor and Remainder Theorem

Question 1

The cubic polynomial \(3x^3+ax^2+bx-12\) has a factor \((x-2)\) and leaves a remainder of -20 when divided by \((x-1)\).

Find the value of a and b.

Hint

Full Solution

Question 2

Given that \(ax^3+bx^2+17x-6\) is exactly divisible by \((x-1)(x-2)\), find the value of a and b.

Hint

Full Solution

Question 3

It is given that \(f(x)=x^3+ax^2+bx+8\)

a. Given that \(x^2-4\) is a factor of \(f(x)\), find the values of a and b.

b. Factorize \(f(x)\) into a product of linear factors.

c. Sketch the graph of \(y=f(x)\) labelling any stationary points and the x and y intercepts.

d. Hence state the range of values of k for which \(f(x)=k\) has exactly one root.

Hint

Full Solution

Sums and Products of Roots

Question 1

The quadratic equation \(3x^{ 2 }−8x+2=0\) has roots \(\alpha\) and \(\beta\).

a. Without solving the equation, find the value of \(\alpha + \beta\) and \(\alpha \beta\).

b. Another quadratic equation \(3x^{ 2 }+bx+c=0\quad ,\quad b,c\in \mathbb{Z}\) has roots \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\). Find the value of b and c.

Hint

Full Solution

Question 2

In the quadratic equation \(px^{ 2 }−45x+25=0\quad ,\quad p\in \mathbb{Z}\) ,one root is two times the other.

Find the value of p.

Hint

Full Solution

Question 3

Let \(f(x)=2x^{ 4 }+x^{ 3 }−14x^{ 2 }+5x+6\quad ,\quad x\in \mathbb{R}\)

a. For the polynomial equation f(x) = 0 , find the value of

  1. the sum of the roots
  2. the product of the roots

b. A new polynomial is defined by g(x) = f(x - 2). Find the sum of the roots of the equation g(x) = 0

Hint

Full Solution

Question 4

Consider the equation \(8x^{ 3 }−42x^{ 2 }+px−27=0.\)

  1. State
    1. the sum of the roots of the equation
    2. the product of the roots of the equation
  2. The roots of this equation are three consecutive terms of a geometric sequence. Taking the roots to be \(\frac{\alpha}{\beta},\alpha,\alpha\beta\) show that one of the roots is \(\frac{3}{2}\)
  3. Solve the equation.
  4. Find the value of p.

Hint

Full Solution

Question 5

One root of the equation z² + bz + c = 0 is 2+3i where \(b,c\in\mathbb{Z}\).

Find the value of b and the value of c.

Hint

Full Solution

Question 6

The equation \(2z^{ 4 }−9z^{ 3 }+pz^{ 2 }+qz−174=0 \quad,\quad p,q\in\mathbb{Z}\) has two real roots \(\alpha\) and \(\beta\) and two complex roots \(\gamma\) and \(\delta\) where \(\gamma=2-5i\).

a. Show that \(\alpha+\beta=\frac{1}{2}.\)

b. Find \(\alpha\beta\).

c. Hence find the two real roots α and β.

d. Find the values of p and q.

Hint

Full Solution

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