Certainly! Let's go through the steps to find [tex]\( f(g(x)) \)[/tex] and [tex]\( g(f(x)) \)[/tex] for the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
The functions are:
[tex]\[ f(x) = \frac{2x + 4}{x + 6} \][/tex]
[tex]\[ g(x) = \frac{6x - 4}{2 - x} \][/tex]
### (a) Find [tex]\( f(g(x)) \)[/tex]
To find [tex]\( f(g(x)) \)[/tex], we need to substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex].
1. Substitute [tex]\( g(x) \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[ f(g(x)) = f \left( \frac{6x - 4}{2 - x} \right) \][/tex]
2. Take the expression for [tex]\( g(x) \)[/tex] and plug it into [tex]\( f(x) \)[/tex]:
[tex]\[ f \left( \frac{6x - 4}{2 - x} \right) = \frac{2 \left( \frac{6x - 4}{2 - x} \right) + 4}{ \left( \frac{6x - 4}{2 - x} \right) + 6 } \][/tex]
3. Simplify the expression step-by-step:
[tex]\[ = \frac{\frac{2(6x - 4)}{2 - x} + 4}{\frac{6x - 4}{2 - x} + 6} \][/tex]
4. Combine terms in the numerator and denominator:
[tex]\[ = \frac{\frac{12x - 8 + 4(2 - x)}{2 - x}}{\frac{6x - 4 + 6(2 - x)}{2 - x}} \][/tex]
[tex]\[ = \frac{\frac{12x - 8 + 8 - 4x}{2 - x}}{\frac{6x - 4 + 12 - 6x}{2 - x}} \][/tex]
[tex]\[ = \frac{\frac{8x}{2 - x}}{\frac{8}{2 - x}} \][/tex]
5. Simplify each fraction by cancelling out the common terms:
[tex]\[ = \frac{8x \div (2 - x)}{8 \div (2 - x)} \][/tex]
6. Further simplification shows that:
[tex]\[ f(g(x)) = x \][/tex]
### (b) Find [tex]\( g(f(x)) \)[/tex]
To find [tex]\( g(f(x)) \)[/tex], we need to substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
1. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[ g(f(x)) = g \left( \frac{2x + 4}{x + 6} \right) \][/tex]
2. Take the expression for [tex]\( f(x) \)[/tex] and plug it into [tex]\( g(x) \)[/tex]:
[tex]\[ g \left( \frac{2x + 4}{x + 6} \right) = \frac{6 \left( \frac{2x + 4}{x + 6} \right) - 4}{2 - \left( \frac{2x + 4}{x + 6} \right)} \][/tex]
3. Simplify the expression step-by-step:
[tex]\[ = \frac{\frac{6(2x + 4)}{x + 6} - 4}{2 - \frac{2x + 4}{x + 6}} \][/tex]
4. Combine terms in the numerator and denominator:
[tex]\[ = \frac{\frac{12x + 24 - 4(x + 6)}{x + 6}}{ \frac{2(x + 6) - (2x + 4)}{x + 6}} \][/tex]
[tex]\[ = \frac{\frac{12x + 24 - 4x - 24}{x + 6}}{\frac{2x + 12 - 2x - 4}{x + 6}} \][/tex]
[tex]\[ = \frac{\frac{8x}{x + 6}}{\frac{8}{x + 6}} \][/tex]
5. Simplify each fraction by cancelling out the common terms:
[tex]\[ = \frac{8x \div (x + 6)}{8 \div (x + 6)} \][/tex]
6. Further simplification shows that:
[tex]\[ g(f(x)) = x \][/tex]
Therefore, the results are:
(a) [tex]\( f(g(x)) = x \)[/tex]
(b) [tex]\( g(f(x)) = x \)[/tex]
