To find the domain of the composite function [tex]\( (f \circ g)(x) \)[/tex], which is [tex]\( f(g(x)) \)[/tex], we need to consider both the domain of [tex]\( g(x) \)[/tex] and the domain of [tex]\( f(x) \)[/tex] when applied to [tex]\( g(x) \)[/tex].
Firstly, let’s find the domain of [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{5}{-2x + 9} \][/tex]
For [tex]\( g(x) \)[/tex] to be defined, the denominator [tex]\(-2x + 9\)[/tex] must not be zero:
[tex]\[ -2x + 9 \neq 0 \][/tex]
Solving for [tex]\( x \)[/tex] when [tex]\( -2x + 9 = 0 \)[/tex]:
[tex]\[ -2x + 9 = 0 \][/tex]
[tex]\[ -2x = -9 \][/tex]
[tex]\[ x = \frac{9}{2} \][/tex]
Therefore, the domain of [tex]\( g(x) \)[/tex] is all real numbers except [tex]\( x = \frac{9}{2} \)[/tex]:
[tex]\[ \text{Domain of } g(x) : \mathbb{R} - \left\{ \frac{9}{2} \right\} \][/tex]
Next, consider the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \frac{1}{x + 5} \][/tex]
For [tex]\( f(x) \)[/tex] to be defined, the denominator [tex]\( x + 5 \)[/tex] must not be zero:
[tex]\[ x + 5 \neq 0 \][/tex]
[tex]\[ x \neq -5 \][/tex]
Now, we are interested in [tex]\( f(g(x)) = f\left( \frac{5}{-2x+9} \right) \)[/tex]:
[tex]\[ f(g(x)) = \frac{1}{g(x) + 5} = \frac{1}{\frac{5}{-2x+9} + 5} \][/tex]
For [tex]\( f(g(x)) \)[/tex] to be defined, the expression inside [tex]\( f \)[/tex] must be well-defined and the argument to [tex]\( f \)[/tex] must also satisfy the condition for [tex]\( f \)[/tex]:
[tex]\[ \frac{5}{-2x + 9} + 5 \neq 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ \frac{5}{-2x + 9} + 5 = 0 \][/tex]
[tex]\[ \frac{5}{-2x + 9} = -5 \][/tex]
[tex]\[ 5 = -5(-2x + 9) \][/tex]
[tex]\[ 5 = 10x - 45 \][/tex]
[tex]\[ 50 = 10x \][/tex]
[tex]\[ x = 5 \][/tex]
Therefore, [tex]\( x = 5 \)[/tex] causes the expression before [tex]\( f \)[/tex] to be zero, which is not allowed. This value [tex]\( x = 5 \)[/tex] must also be excluded from the domain.
Now, combining these restrictions, we have two critical points to exclude from the domain:
- [tex]\( x = \frac{9}{2} \)[/tex]
- [tex]\( x = 5 \)[/tex]
Thus, the domain of [tex]\( (f \circ g)(x) \)[/tex] is all real numbers except [tex]\( x = \frac{9}{2} \)[/tex] and [tex]\( x = 5 \)[/tex]:
[tex]\[ \text{Domain of } (f \circ g)(x) : \mathbb{R} - \left\{ \frac{9}{2}, 5 \right\} \][/tex]
