Answer ALL questions - Edexcel - A-Level Maths Pure - Question 3 - 2018 - Paper 2
Question 3
Answer ALL questions. Write your answers in the spaces provided.
g(x) = \frac{2x + 5}{x - 3} \, \text{ for } \, x > 5
(a) Find gg(5).
(b) State the range of g.
(... show full transcript
Worked Solution & Example Answer:Answer ALL questions - Edexcel - A-Level Maths Pure - Question 3 - 2018 - Paper 2
Step 1
Find gg(5).
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Answer
To find gg(5), we first need to calculate g(5).
Calculate g(5):
g(5)=5−32(5)+5=210+5=215
Next, substitute g(5) into g:
gg(5)=g(215)=215−32(215)+5
Simplify:
gg(5)=215−2615+5=2920=920×2=940.
Thus, the final answer is (gg(5) = \frac{40}{9}).
Step 2
State the range of g.
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Answer
The function g(x) approaches a vertical asymptote at x = 3 and is undefined for x = 3. The range of g is all real numbers greater than 2 because:
As x approaches 3 from the left, g(x) tends towards ( -\infty ).
As x approaches ( \infty ), g(x) approaches the horizontal asymptote of 2.
Thus, the range is ( 2 < y < \frac{15}{2} ), which can also be stated as ( g(x) > 2 ).
Step 3
Find g^{-1}(x), stating its domain.
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Answer
To find the inverse of g(x), first rewrite the function:
y=g(x)=x−32x+5.
Interchange x and y:
x=y−32y+5.
Cross-multiply:
x(y−3)=2y+5.
Rearrange to isolate y:
xy−3x=2y+5⇒xy−2y=3x+5⇒y(x−2)=3x+5.
Solve for y:
y=x−23x+5.
Therefore, ( g^{-1}(x) = \frac{3x + 5}{x - 2} ) with a domain of ( x \neq 2 ).
