Question 1
Given that f(x)=x-9, find:
1(a):
f^{-1}(x)ANSWER: Multiple Choice (Type 1)
A: 9x
B: \dfrac{x}{9}
C: x+9
D: 9-x
Answer: C
Workings:
In the original equation we subtract 9 from x
So for the inverse we do the inverse of subtracting and add 9 to x
This gives f^{-1}(x)=x+9
Marks = 1
1(b):
f^{-1}(4)ANSWER: Simple Text Answer
Answer: 13
Workings:
Substituting x=9 into our equation for f^{-1}(x) gives
f^{-1}(4)=4+9=13
Marks = 1
Question 2
Given that f(x)=5x-3, find:
2(a):
f^{-1}(x)ANSWER: Multiple Choice Answer (Type 1)
A: 3-5x
B: 3x-5
C:\dfrac{5+x}{3}
D:\dfrac{x+3}{5}
Answer: D
Workings:
To get from x to 5x-3 we would multiply by 5, then subtract 3.
Doing the inverse, we would therefore add 3 and divide by 5.
So, adding 3 gives us x+3. Dividing this by 5 will then give us f^{-1}(x)=\dfrac{x+3}{5}.
Marks = 1
2(b):
f^{-1}(3)ANSWER: Fraction
Answer: \dfrac{6}{5}
Workings:
Substituting x=\dfrac{6}{5} into the equation for f^{-1}(x) gives
f^{-1}(3)=\dfrac{3+3}{5}=\dfrac{6}{5}
Marks = 1
Question 3
Given that f(x)=\dfrac{x-8}{3}, find:
3(a):
f^{-1}(x)ANSWER: Multiple Choice (Type 1)
A: 3x-8
B: 3x+8
C: 8x+3
D: 8x-3
Answer: B
Workings:
To get from x to \dfrac{x-8}{3} we would subtract 8 and divide by 3
The inverse of this would mean multiplying by 3 and adding 8.
This gives us f^{-1}(x)=3x+8
Marks = 1
3(b):
f^{-1}(10)ANSWER: Simple Text Answer
Answer: 38
Workings:
Substituting x =10 into f^{-1}(x) gives
f^{-1}(10)=3\times 10+8=38
Marks = 1
Question 4
Given that f(x)=\sqrt{\dfrac{-x+2}{4}}, find:
4(a):
f^{-1}(x)ANSWER: Multiple Choice (Type 1)
A: 4x-2
B: \dfrac{(x+2)^2}{16}
C: 4x^2-2
D: 2-4x^2
Answer: D
Workings:
To get from x to \sqrt{\dfrac{-x+2}{4}} we would need to multiply by -1, then add 2, divide by 4 and finally square root.
To get the inverse we need to do this the other way round doing the inverse of each operation.
First we would square x to get x^2.
We then multiply by 4 to get 4x^2.
Subtract 2 to get 4x^2-2.
Finally divide/multiply by -1 to get 2-4x^2.
Marks = 2
4(b):
f^{-1}(3)ANSWER: Simple Text Answer
Answer: -34
Workings:
Substitute x=3 into the equation for f^{-1}(x)
This gives f^{-1}(3)=2-4(3)^2=-34
Marks = 2
Question 5
Functions f and g are defined by f(x) = 2x+4 and g(x) = 3x+1.
5(a):
Find the value of x when f(x) = g(x).
ANSWER: Simple Text Answer
Answer: x=3
Workings:
If f(x)=g(x) then 2x+4=3x+1
Rearranging gives x=3.
Marks = 1
5(b):
Find and simplify the expression for fg(x)
ANSWER: Multiple Choice (Type 1)
A: 6x+6
B: 6x+13
C: 5x+5
D: 6x^2+14x+4
Answer: A
Workings:
This is a composite function with g as a function of f.
This means we substitute the equation for g(x) into the equation for f(x) which gives f(3x+1)
fg(x)=f(3x+1)=2(3x+1)+4=6x+2+4=6x+6
Marks = 2
5(c):
Find and simplify the expression for gf(x)
ANSWER: Multiple Choice (Type 1)
A: 7x+4
B: 5x+5
C: 6x+13
D: 8x+11
Answer: C
Workings:
We are now doing f as a function of g, so substituting the equation for f(x) into the equation for g(x).
This gives g(2x+4)
gf(x)=g(2x+4)=3(2x+4)+1=6x+12+1=6x+13
Marks = 2
Question 6
Given that f(x) = \dfrac{5}{x-1} and g(x) = 4-2x, find:
6(a):
ff^{-1}(-2)ANSWER: Simple Text Answer
Answer:-2
Workings:
We get from x to f(x) by subtracting 1, putting it to the power of -1 and multiplying by 5.
Doing the inverse of this means dividing by 5, putting it to the power of -1 again and adding 1.
This gives us f^{-1}(x)=\dfrac{5}{x}+1.
To find f^{-1}(-2) we need to substitute x=-2 into the equation for f^{-1}(x).
This gives f^{-1}(-2)=\dfrac{5}{-2}+1=-\dfrac{3}{2}
To find ff^{-1}(2) we find f(-\dfrac{3}{2})
ff^{-1}(-2)=f(-\dfrac{3}{2})=\dfrac{5}{-\dfrac{3}{2}-1}=-2
Marks = 2
6(b):
gg(-3)ANSWER: Simple Text Answer
Answer: -16
Workings:
g(-3)=4-2(-3)=10
This gives us gg(-3)=g(10)
g(10)=4-2\times 10 = -16
Marks = 2
6(c):
A simplified expression for fg(x)
ANSWER: Multiple Choice (Type 1)
A: \dfrac{3-2x}{5}
B: \dfrac{5}{3-2x}
C: 4-\dfrac{10}{x-1}
D: \dfrac{20-10x}{x-1}
Answer: B
Workings:
fg(x)=f(4-2x)
f(4-2x)=\dfrac{5}{4-2x-1}=\dfrac{5}{3-2x}
Marks = 2
Question 7
7(a):
Given that f(x) = x^2 – a and g(x) = x+b, find an expression for fg(x) in terms of a and b.
ANSWER: Multiple Choice (Type 1)
A: x^2-a+b
B: x^2+x+b-a
C: x^3+bx^2-ax-ab
D: (x+b)^2-a
Answer: D
Workings:
fg(x)=f(x+b)
f(x+b)=(x+b)^2-a
Marks = 2
7(b):
If b=2a, and a= -3, what is the value of fg(5)?
ANSWER: Simple Text Answer
Answer: 4
Workings:
To find fg(5) we can substitute x=5 into our equation answer for the previous question.
fg(5)=(5+b)^2-a
a=-3 and by deduction b=-6
Substituting these values into our equation for fg(5) gives fg(5)=(5-6)^2-(-3)=4
Marks = 2