The functions f and g are such that
f(x) = 3x - 1 and g(x) = x² + 4
(a) Find f^{-1}(x)
f^{-1}(x) = ______________
Given that fg(x) = 2g(f(x)),
(b) show that 15x² - 12x - 1 = 0 - Edexcel - GCSE Maths - Question 22 - 2019 - Paper 1
Question 22
The functions f and g are such that
f(x) = 3x - 1 and g(x) = x² + 4
(a) Find f^{-1}(x)
f^{-1}(x) = ______________
Given that fg(x) = 2g(f(x)),
(b) show that 15x²... show full transcript
Worked Solution & Example Answer:The functions f and g are such that
f(x) = 3x - 1 and g(x) = x² + 4
(a) Find f^{-1}(x)
f^{-1}(x) = ______________
Given that fg(x) = 2g(f(x)),
(b) show that 15x² - 12x - 1 = 0 - Edexcel - GCSE Maths - Question 22 - 2019 - Paper 1
Step 1
Find f^{-1}(x)
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Answer
To find the inverse of the function f(x) = 3x - 1, we will follow these steps:
Set f(x) equal to y:
Let y = 3x - 1.
Solve for x in terms of y:
Rearranging the equation gives:
y+1=3x
Therefore, x=3y+1.
Express the inverse function:
Therefore, the inverse function is:
f−1(x)=3x+1.
Step 2
show that 15x² - 12x - 1 = 0
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Answer
Given that fg(x) = 2g(f(x)), we can express this as follows:
Calculate fg(x):
Substitute g(x) into f(x):
fg(x)=f(g(x))=f(x2+4)=3(x2+4)−1=3x2+12−1=3x2+11.
Calculate g(f(x)):
Substitute f(x) into g(x):
g(f(x))=g(3x−1)=(3x−1)2+4=9x2−6x+1+4=9x2−6x+5.
Set the equation using fg(x) and g(f(x)):
Thus, we have:
fg(x)=3x2+11=2g(f(x))=2(9x2−6x+5)=18x2−12x+10.
Rearranging the equation:
3x2+11=18x2−12x+10
leads to 0=18x2−12x+10−3x2−11
simplifying to: 15x2−12x−1=0.
Therefore, we have shown that the equation holds.
