1. Two functions f(x) = 3 - 3x and g(x) = x
(a) Find the value of g(-3).
(b) Find f'(x).
2. f(x) = 8 - 3x and g(x) = 10/(x+1)
(a) Write down the value of x for which g(x) is undefined.
(b) Find f(8/3).
3. Functions f and g are given, such that f(x) = 3 - 4x and g(x) = x
(a) Find gf(2).
(b) Solve the equation f(x) = 5 - x.
4. It is given that p(x) = 4x - 7 and q(x) = x
(a) Find p(-2).
(b) Find x when q(x) = 2197.
Answer (2)
I provided step-by-step solutions for each part of the functions given in the questions, calculating values, derivatives, and solving equations. The results include specific values from function evaluations and the derivation of functions, ensuring clarity and correctness throughout. Each step followed logical mathematical processes to arrive at the respective conclusions.
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Let's solve each part step-by-step for all four items:
Two functions f ( x ) = 3 − 3 x and g ( x ) = x 2 are given.
(a) Find the value of g ( − 3 ) .
g ( − 3 ) = ( − 3 ) 2 = 9.
(b) Find f ′ ( x ) .
The function is f ( x ) = 3 − 3 x . To find the derivative f ′ ( x ) , recall that the derivative of a x n with respect to x is n ⋅ a x n − 1 . Here, f ( x ) is a linear function, so:
f ′ ( x ) = d x d ( 3 − 3 x ) = 0 − 3 = − 3.
f ( x ) = 8 − 3 x and g ( x ) = x + 1 10
(a) Write down the value of x for which g ( x ) is undefined.
g ( x ) is undefined when the denominator is zero. So, solve:
x + 1 = 0 x = − 1.
(b) Find f ( 3 8 ) .
Substitute x = 3 8 into f ( x ) :
f ( 3 8 ) = 8 − 3 ( 3 8 ) = 8 − 8 = 0.
Functions f and g are given, such that f ( x ) = 3 − 4 x and g ( x ) = x 2 where x is a real number.
(a) Find g f ( 2 ) .
First, find f ( 2 ) :
f ( 2 ) = 3 − 4 ( 2 ) = 3 − 8 = − 5.
Now find g ( f ( 2 )) = g ( − 5 ) :
g ( − 5 ) = ( − 5 ) 2 = 25.
(b) Solve the equation f ( x ) = 5 − x .
Set 3 − 4 x = 5 − x and solve for x :
3 − 4 x = 5 − x 3 x = 2 x = 3 2 .
It is given that p ( x ) = 4 x − 7 and q ( x ) = x 3 .
(a) Find p ( − 2 ) .
Substitute x = − 2 into p ( x ) :
p ( − 2 ) = 4 ( − 2 ) − 7 = − 8 − 7 = − 15.
(b) Find x when q ( x ) = 2197 .
Solve x 3 = 2197 :
Since 2197 = 1 3 3 , we have x = 13 .
