Sure, let's tackle this problem step by step.
We have the two functions:
[tex]\[ f(x) = 2x - 3 \][/tex]
[tex]\[ g(x) = \frac{1}{2}(x + 3) \][/tex]
We are asked to find the compositions [tex]\((f \circ g)(x)\)[/tex] and [tex]\((g \circ f)(x)\)[/tex] and check if they are equal.
### Finding [tex]\((f \circ g)(x)\)[/tex]
The composition [tex]\((f \circ g)(x)\)[/tex] means applying [tex]\(g(x)\)[/tex] first and then applying [tex]\(f\)[/tex] to the result of [tex]\(g(x)\)[/tex]:
[tex]\[ (f \circ g)(x) = f(g(x)) \][/tex]
1. Compute [tex]\(g(x)\)[/tex]:
[tex]\[ g(x) = \frac{1}{2}(x + 3) \][/tex]
2. Substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f\left( \frac{1}{2}(x + 3) \right) \][/tex]
[tex]\[ = 2 \left( \frac{1}{2}(x + 3) \right) - 3 \][/tex]
3. Simplify the expression:
[tex]\[ = 2 \cdot \frac{1}{2}(x + 3) - 3 \][/tex]
[tex]\[ = (x + 3) - 3 \][/tex]
[tex]\[ = x \][/tex]
Thus, [tex]\((f \circ g)(x) = x\)[/tex].
### Finding [tex]\((g \circ f)(x)\)[/tex]
The composition [tex]\((g \circ f)(x)\)[/tex] means applying [tex]\(f(x)\)[/tex] first and then applying [tex]\(g\)[/tex] to the result of [tex]\(f(x)\)[/tex]:
[tex]\[ (g \circ f)(x) = g(f(x)) \][/tex]
1. Compute [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = 2x - 3 \][/tex]
2. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[ g(f(x)) = g(2x - 3) \][/tex]
[tex]\[ = \frac{1}{2}((2x - 3) + 3) \][/tex]
3. Simplify the expression:
[tex]\[ = \frac{1}{2}(2x - 3 + 3) \][/tex]
[tex]\[ = \frac{1}{2}(2x) \][/tex]
[tex]\[ = x \][/tex]
Thus, [tex]\((g \circ f)(x) = x\)[/tex].
### Determining Equality
We have:
[tex]\[ (f \circ g)(x) = x \][/tex]
[tex]\[ (g \circ f)(x) = x \][/tex]
Since both compositions yield [tex]\(x\)[/tex], we can conclude:
[tex]\[ (f \circ g)(x) = (g \circ f)(x) = x \][/tex]
Therefore, [tex]\((f \circ g)(x) = (g \circ f)(x)\)[/tex].
