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Consider the functions f and g where f (x) = 3x − 5 and g (x) = x − 2 . (a) Find the inverse function, f^−1 . (b) Given that g^−1(x) = x + 2 , find (g^−1 o f )(x) . (c) Given also that (f^−1 o g)(x) = (x + 3)/3 , solve (f^−1 o g)(x) = (g^−1 o f)(x)

(a) Start with f(x)= 3x − 5; y=3x - 5, and switch x and y before rearanging to get y in terms of x again: y=3x - 5 x=3y - 5 (x+5)/3=y Therefore f^−1(x)=(x+5)/3 (b) Start with f(x)= 3x − 5 and and sub result ...
IB
Answered by Isobel B. • Maths tutor
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Solve the differential equation: dy/dx = tan^3(x)sec^2(x)

dy/dx = tan 3 (x)sec 2 (x) Integrate both sides ==> ∫dy= ∫ tan 3 (x)sec 2 (x) dx Use the substitution u=tan(x) And by differentiation du/dx = sec 2 (x) , which leads to dx = du/sec 2 (x) ==> and subbin...
RS
Answered by Ryan S. • Maths tutor
12882 Views

Differentiate x^(1/2)ln(3x) with respect to x.

First we notice that this is a product of two functions of x, so we are going to use the product rule. Recall (uv)'(x)=u'(x)v(x)+v'(x)u(x). Let u(x)=x^(1/2) and v(x)=ln(3x). We need to find u'(x) and v'(x). ...
AR
Answered by Aidan R. • Maths tutor
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Identify the stationary points of f(x)=3x^3+2x^2+4 (by finding the first and second derivative) and determine their nature.

f'(x)=9x 2 ​+4x, and f''(x)=18x+4 (derivatives) f'(x)=0 at x=0 or x=-4/9 when x=0 f''(x)>0 therefore a minimum value, when x=-4/9 f''(x)<0 and thus a maximum value.
SO
Answered by Sieff O. • Maths tutor
4250 Views

Find values of x for which 2x^2 < 5x + 12

Start by rearranging the inequality - make sure the sign next to the x 2 term is positive to make it easier: 2x 2 -5x - 12&lt; 0 Next step is to factorise this quadratic. To do this, remember that the genera...
SB
Answered by Sean B. • Maths tutor
15879 Views