Question
f(x)=4x-1 \( g(x)=x^{2} \) \(h(x)=3^{-x}\)
(a) Find in its simplest form
(i) f(x-3),
………………………………………….
(ii) g(5x).
………………………………………….
(b) Find \(f^{-1}(x).\)
\(f^{-1}(x)\)…………………………………..
(c) Find the value of hh(l) , correct to 4 significant figures.
………………………………………….
(d) (i) Show that g(3-2x)-h(-3) can be written as \(9x^{2}-12x-23.\)
(ii) Use the quadratic formula to solve \(9x^{2}-12x-23.\)
Give your answers correct to 2 decimal places.
x = ………………………. or x = ……………………….
(e) Find x when f(61)=h(x).
x = ………………………………………….
Answer/Explanation
(a))(i) 4x- 13 final answer
(ii)\( 25x^{2}\) final answer
(b)\(\frac{x+4}{4}\) or \(\frac{x}{4}+\frac{1}{4}\)
(c) 0.6934 final answer
(d)(i)\((3x-2)^{2}-3^{-(-3)}\)
\(9x^{2}-6x-6x+4-27 \)or
\(9x^{2}-12x+4-27\)
leading to \(9x^{2}-12x-23\)
(ii)\(\frac{-(-12)\pm \sqrt{(-12)^{2}-4(9)(-23)}}{2\times 9}\)
or better
-1.07,2.40 final answers
(e) -5 final answer
Question
\(f(x)=1+4x \) \( g(x)=x^{2}\)
Find
(i) gf(3),
(ii) fg(x),
(iii) \(f^{-1}\) ( f)x
(b) Find the value of x when f(x)=15
Answer/Explanation
3(a)(i) 169
3(a)(ii) \( 1+4x^{2}\) final answer
3(a)(iii) x
3(b) \(3.5 or \frac{7}{2}\)
Question
\(f\left ( x \right )=\frac{3}{x+2},x\neq -2\) \(g\left ( x \right )=8x-5\) \(h\left ( x \right )=x^{2}+6\)
(a) Work out \(g\left ( \frac{1}{4} \right )\).[1]
(b) Work out ff(2).[2]
(c) Find gg(x), giving your answer in its simplest form.[2]
(d) Find \(g^{-1}\left ( x \right )\).
\(g^{-1}\left ( x \right )=\)[2]
(e) Write \(g\left ( x \right )-f\left ( x \right )\) as a single fraction in its simplest form.[3]
(f) (i) Show that \(hg\left ( x \right )=19\) simplifies to \(16x^{2}-20x+3=0\).[3]
(ii) Use the quadratic formula to solve \(16x^{2}-20x+3=0\).
Show all your working and give your answers correct to 2 decimal places.
x = ___ or x = ___ [4]
Answer/Explanation
Ans:
8(a) −3
8(b) \(\frac{12}{11}oe\)
8(c) 64x − 45 final answer
8(d) \(\frac{x+5}{8}oe\) final answer
8(e) \(\frac{8x^{2}+11x-13}{x+2}\) final answer
8(f)(i) \(\left ( 8x-5 \right )^{2}+6=19\)
\(64^{2}-40x-40x+25\)
\(64^{2}-40x-40x+25+6=19\) oe
leading to \(16^{2}-20x+3=0\)
8(f)(ii) \(\frac{\left [ — \right ]20\pm \sqrt{\left ( \left [ – \right ] 20\right )^{2}-4\left ( 16 \right )\left ( 3 \right )}}{2×16}oe\)
0.17 and 1.08 final ans