Answer :
To determine if two functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are inverses of each other, we need to check two primary conditions:
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's proceed with each step carefully.
### Step 1: Check [tex]\( f(g(x)) \)[/tex]
Given:
[tex]\[ f(x) = 2x - 6 \][/tex]
[tex]\[ g(x) = \frac{x}{2} + 3 \][/tex]
First, compute [tex]\( f(g(x)) \)[/tex]:
[tex]\[ g(x) = \frac{x}{2} + 3 \][/tex]
Now, substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{x}{2} + 3\right) \][/tex]
[tex]\[ = 2\left(\frac{x}{2} + 3\right) - 6 \][/tex]
[tex]\[ = 2 \cdot \frac{x}{2} + 2 \cdot 3 - 6 \][/tex]
[tex]\[ = x + 6 - 6 \][/tex]
[tex]\[ = x \][/tex]
So, we have:
[tex]\[ f(g(x)) = x \][/tex]
### Step 2: Check [tex]\( g(f(x)) \)[/tex]
Next, compute [tex]\( g(f(x)) \)[/tex]:
[tex]\[ f(x) = 2x - 6 \][/tex]
Now, substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(2x - 6) \][/tex]
[tex]\[ = \frac{2x - 6}{2} + 3 \][/tex]
[tex]\[ = \frac{2x}{2} - \frac{6}{2} + 3 \][/tex]
[tex]\[ = x - 3 + 3 \][/tex]
[tex]\[ = x \][/tex]
So, we have:
[tex]\[ g(f(x)) = x \][/tex]
### Final Conclusion
Since both conditions:
[tex]\[ f(g(x)) = x \][/tex]
[tex]\[ g(f(x)) = x \][/tex]
are satisfied, [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
Therefore, the answer is:
[tex]\[ \boxed{\text{Yes}} \][/tex]
1. [tex]\( f(g(x)) = x \)[/tex]
2. [tex]\( g(f(x)) = x \)[/tex]
Let's proceed with each step carefully.
### Step 1: Check [tex]\( f(g(x)) \)[/tex]
Given:
[tex]\[ f(x) = 2x - 6 \][/tex]
[tex]\[ g(x) = \frac{x}{2} + 3 \][/tex]
First, compute [tex]\( f(g(x)) \)[/tex]:
[tex]\[ g(x) = \frac{x}{2} + 3 \][/tex]
Now, substitute [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = f\left(\frac{x}{2} + 3\right) \][/tex]
[tex]\[ = 2\left(\frac{x}{2} + 3\right) - 6 \][/tex]
[tex]\[ = 2 \cdot \frac{x}{2} + 2 \cdot 3 - 6 \][/tex]
[tex]\[ = x + 6 - 6 \][/tex]
[tex]\[ = x \][/tex]
So, we have:
[tex]\[ f(g(x)) = x \][/tex]
### Step 2: Check [tex]\( g(f(x)) \)[/tex]
Next, compute [tex]\( g(f(x)) \)[/tex]:
[tex]\[ f(x) = 2x - 6 \][/tex]
Now, substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(2x - 6) \][/tex]
[tex]\[ = \frac{2x - 6}{2} + 3 \][/tex]
[tex]\[ = \frac{2x}{2} - \frac{6}{2} + 3 \][/tex]
[tex]\[ = x - 3 + 3 \][/tex]
[tex]\[ = x \][/tex]
So, we have:
[tex]\[ g(f(x)) = x \][/tex]
### Final Conclusion
Since both conditions:
[tex]\[ f(g(x)) = x \][/tex]
[tex]\[ g(f(x)) = x \][/tex]
are satisfied, [tex]\( f \)[/tex] and [tex]\( g \)[/tex] are indeed inverses of each other.
Therefore, the answer is:
[tex]\[ \boxed{\text{Yes}} \][/tex]
