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Sagot :
To analyze which function has the same domain as the composite function [tex]\( (m \circ n)(x) \)[/tex], let's go through the solution step by step:
1. Define the Functions:
- [tex]\( m(x) = \frac{x + 5}{x - 1} \)[/tex]
- [tex]\( n(x) = x - 3 \)[/tex]
2. Composite Function [tex]\( (m \circ n)(x) \)[/tex]:
- The composite function [tex]\( (m \circ n)(x) \)[/tex] means [tex]\( m(n(x)) \)[/tex].
- Substitute [tex]\( n(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ m(n(x)) = m(x - 3) = \frac{(x - 3) + 5}{(x - 3) - 1} = \frac{x - 3 + 5}{x - 3 - 1} = \frac{x + 2}{x - 4} \][/tex]
3. Determine the Domain of [tex]\( (m \circ n)(x) \)[/tex]:
- Identify where [tex]\( (m \circ n)(x) \)[/tex] is undefined:
[tex]\[ \text{The function } \frac{x + 2}{x - 4} \text{ is undefined when its denominator is } 0 \][/tex]
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]
- Therefore, [tex]\( (m \circ n)(x) \)[/tex] is undefined for [tex]\( x = 4 \)[/tex]. The domain of [tex]\( (m \circ n)(x) \)[/tex] is all real numbers except [tex]\( x = 4 \)[/tex].
4. Compare Potential Functions [tex]\( h(x) \)[/tex]:
- Check each option to see which function is undefined at [tex]\( x = 4 \)[/tex]:
- [tex]\( h(x) = \frac{x+5}{11} \)[/tex]:
[tex]\[ \text{This function is defined for all } x \text{ since there are no restrictions on the domain}. \][/tex]
- [tex]\( h(x) = \frac{11}{x-1} \)[/tex]:
[tex]\[ \text{This function is undefined for } x = 1 \text{ (since } x = 1 \text{ would make the denominator 0).} \][/tex]
- [tex]\( h(x) = \frac{11}{x-4} \)[/tex]:
[tex]\[ \text{This function is undefined for } x = 4 \text{ (since } x = 4 \text{ would make the denominator 0).} \][/tex]
- [tex]\( h(x) = \frac{11}{x-3} \)[/tex]:
[tex]\[ \text{This function is undefined for } x = 3 \text{ (since } x = 3 \text{ would make the denominator 0).} \][/tex]
5. Conclusion:
- The function [tex]\( h(x) = \frac{11}{x-4} \)[/tex] has the same domain as [tex]\( (m \circ n)(x) \)[/tex], which is all real numbers except [tex]\( x = 4 \)[/tex].
Therefore, the correct function [tex]\( h(x) \)[/tex] with the same domain as [tex]\( (m \circ n)(x) \)[/tex] is:
[tex]\[ h(x) = \frac{11}{x-4} \][/tex]
1. Define the Functions:
- [tex]\( m(x) = \frac{x + 5}{x - 1} \)[/tex]
- [tex]\( n(x) = x - 3 \)[/tex]
2. Composite Function [tex]\( (m \circ n)(x) \)[/tex]:
- The composite function [tex]\( (m \circ n)(x) \)[/tex] means [tex]\( m(n(x)) \)[/tex].
- Substitute [tex]\( n(x) \)[/tex] into [tex]\( m(x) \)[/tex]:
[tex]\[ m(n(x)) = m(x - 3) = \frac{(x - 3) + 5}{(x - 3) - 1} = \frac{x - 3 + 5}{x - 3 - 1} = \frac{x + 2}{x - 4} \][/tex]
3. Determine the Domain of [tex]\( (m \circ n)(x) \)[/tex]:
- Identify where [tex]\( (m \circ n)(x) \)[/tex] is undefined:
[tex]\[ \text{The function } \frac{x + 2}{x - 4} \text{ is undefined when its denominator is } 0 \][/tex]
[tex]\[ x - 4 = 0 \implies x = 4 \][/tex]
- Therefore, [tex]\( (m \circ n)(x) \)[/tex] is undefined for [tex]\( x = 4 \)[/tex]. The domain of [tex]\( (m \circ n)(x) \)[/tex] is all real numbers except [tex]\( x = 4 \)[/tex].
4. Compare Potential Functions [tex]\( h(x) \)[/tex]:
- Check each option to see which function is undefined at [tex]\( x = 4 \)[/tex]:
- [tex]\( h(x) = \frac{x+5}{11} \)[/tex]:
[tex]\[ \text{This function is defined for all } x \text{ since there are no restrictions on the domain}. \][/tex]
- [tex]\( h(x) = \frac{11}{x-1} \)[/tex]:
[tex]\[ \text{This function is undefined for } x = 1 \text{ (since } x = 1 \text{ would make the denominator 0).} \][/tex]
- [tex]\( h(x) = \frac{11}{x-4} \)[/tex]:
[tex]\[ \text{This function is undefined for } x = 4 \text{ (since } x = 4 \text{ would make the denominator 0).} \][/tex]
- [tex]\( h(x) = \frac{11}{x-3} \)[/tex]:
[tex]\[ \text{This function is undefined for } x = 3 \text{ (since } x = 3 \text{ would make the denominator 0).} \][/tex]
5. Conclusion:
- The function [tex]\( h(x) = \frac{11}{x-4} \)[/tex] has the same domain as [tex]\( (m \circ n)(x) \)[/tex], which is all real numbers except [tex]\( x = 4 \)[/tex].
Therefore, the correct function [tex]\( h(x) \)[/tex] with the same domain as [tex]\( (m \circ n)(x) \)[/tex] is:
[tex]\[ h(x) = \frac{11}{x-4} \][/tex]
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