Question

F(x) = 3x+1 for x ∈ R

g(x) = 10 ÷ (2 - x) for x not equal to 2
solve the equation gf(x) = 5

Answer 1

To solve the composed equation gf(x) = 5 with given functions f(x) = 3x + 1 and g(x) = 10 ÷ (2 - x), we need to plug f(x) into g and solve for x, which gives us the solution x = -⅓.

Step-by-step explanation:

To solve the equation gf(x) = 5, we must understand that we are looking for the value of x such that when we apply the function g to the result of the function f at x, we get 5. The functions given are f(x) = 3x + 1 for all real numbers x, and g(x) = 10 ÷ (2 - x) for x not equal to 2.

First, we find f(x) and then apply g to that result:

1. Find f(x): f(x) = 3x + 1.
2. Apply g to f(x): g(f(x)) = g(3x + 1) = 10 ÷ (2 - (3x + 1)).
3. Solve for x when g(f(x)) equals 5: 10 ÷ (2 - (3x + 1)) = 5.
4. Rewrite the equation to isolate x: 2 - (3x + 1) = 10 ÷ 5.
5. Simplify to find x: x = -⅓.

Therefore, the solution to the equation gf(x) = 5 is x = -⅓.