Question

1. Two functions f(x) ang g(x) are defined on the set of real numbers by f(x) = 3x² − 2 and g(x) = x + 3. Find (a) f(- 2). (b) g-1 (-³/4). (c) the value of x for which f(g(x)) = g(f(x) ​

Answer 1

(a) 10

(b) -3 3/4

(c) -4/3

Explanation:

Function values can be found by direct substitution of the argument into the function expression. Inverse function values can be found by solving the function equation for the variable value.

(a)

f(-2)= 3(-2)² -2 = 12 -2

f(-2) = 10

__

(b)

g^-1(-3/4)

-3/4 = x +3

-3 3/4 = x . . . . . . subtract 3

__

(c)

Substitute the expressions as required and solve for x.

f(g(x)) = g(f(x))

f(x+3) = g(3x² -2)

3(x +3)² -2 = (3x² -2) +3

3(x²2 +6x +9) -2 = 3x² +1

3x² +18x +25 = 3x² +1

18x = -24

x = -24/18 = -4/3

_____

Additional comment

When solving an equation graphically, we like to write it in the form h(x) = 0 when possible. The graphing calculator easily identifies x-intercepts. This form makes each solution an x-intercept.

f(g(x)) = g(f(x)) ⇔ f(g(x)) -g(f(x)) = 0

__

The value of the inverse function can be found similarly. We want to find x such that g(x) = -3/4, so we can look for the solution to g(x) +3/4 = 0.