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Answer :
To solve the question of which expression is equivalent to [tex]\((f g)(5)\)[/tex], it's important to understand what [tex]\((f g)(5)\)[/tex] represents. Typically, in mathematics, when we see an expression like [tex]\((f g)(x)\)[/tex], it means we're looking at the product of the two functions evaluated at [tex]\(x\)[/tex].
Here's how you can think of it step-by-step:
1. Understand the Notation:
- The notation [tex]\((f g)(5)\)[/tex] usually indicates that we should evaluate both functions, [tex]\(f\)[/tex] and [tex]\(g\)[/tex], at the input value, which is 5 in this case, and then multiply the results. In mathematical terms, this is expressed as [tex]\(f(5) \times g(5)\)[/tex].
2. Compare with Given Options:
- [tex]\(f(5) \times g(5)\)[/tex]: This expression perfectly matches the operation implied by [tex]\((f g)(5)\)[/tex], which is multiplying the values of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] when each is evaluated at 5.
- [tex]\(f(5) + g(5)\)[/tex]: This choice involves adding the results of the two functions evaluated at 5, which doesn't correspond to the operation described by [tex]\((f g)(5)\)[/tex].
- 57(5): This option doesn't logically fit [tex]\((f g)(x)\)[/tex] as it's not related to the multiplication of two function values.
- [tex]\(5 g(5)\)[/tex]: This suggests multiplying 5 by [tex]\(g(5)\)[/tex], which does not correctly represent the operation [tex]\((f g)(5)\)[/tex].
3. Conclusion:
- The correct expression, based on the standard interpretation of [tex]\((f g)(x)\)[/tex] as the product of functions evaluated at a point, is [tex]\(f(5) \times g(5)\)[/tex].
Therefore, the expression that is equivalent to [tex]\((f g)(5)\)[/tex] is [tex]\(f(5) \times g(5)\)[/tex].
Here's how you can think of it step-by-step:
1. Understand the Notation:
- The notation [tex]\((f g)(5)\)[/tex] usually indicates that we should evaluate both functions, [tex]\(f\)[/tex] and [tex]\(g\)[/tex], at the input value, which is 5 in this case, and then multiply the results. In mathematical terms, this is expressed as [tex]\(f(5) \times g(5)\)[/tex].
2. Compare with Given Options:
- [tex]\(f(5) \times g(5)\)[/tex]: This expression perfectly matches the operation implied by [tex]\((f g)(5)\)[/tex], which is multiplying the values of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] when each is evaluated at 5.
- [tex]\(f(5) + g(5)\)[/tex]: This choice involves adding the results of the two functions evaluated at 5, which doesn't correspond to the operation described by [tex]\((f g)(5)\)[/tex].
- 57(5): This option doesn't logically fit [tex]\((f g)(x)\)[/tex] as it's not related to the multiplication of two function values.
- [tex]\(5 g(5)\)[/tex]: This suggests multiplying 5 by [tex]\(g(5)\)[/tex], which does not correctly represent the operation [tex]\((f g)(5)\)[/tex].
3. Conclusion:
- The correct expression, based on the standard interpretation of [tex]\((f g)(x)\)[/tex] as the product of functions evaluated at a point, is [tex]\(f(5) \times g(5)\)[/tex].
Therefore, the expression that is equivalent to [tex]\((f g)(5)\)[/tex] is [tex]\(f(5) \times g(5)\)[/tex].
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