Which Of The Following Expressions Is Equivalent To $(s \cdot T)(6)$?
When dealing with mathematical expressions, particularly those involving functions, it's crucial to understand the notations and how operations apply. Let's thoroughly examine the expression $(s \cdot t)(6)$ and the provided options to identify the equivalent one.
Understanding the Notation
The expression $(s \cdot t)(6)$ represents the product of two functions, s and t, evaluated at the value 6. In mathematical terms, this means we first multiply the functions s and t together, and then we evaluate the resulting function at x = 6. The dot notation $s \cdot t$ signifies pointwise multiplication of the functions. To break it down:
- s and t are functions, meaning they take an input (in this case, x) and produce an output.
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s(x)$ represents the output of function *s* when the input is *x*.
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t(x)$ represents the output of function *t* when the input is *x*.
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s(x) \cdot t(x)$ means we multiply the output of *s* at *x* by the output of *t* at *x*. This gives us a new function, which is the product of *s* and *t*.
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(s \cdot t)(6)$ means we evaluate the new function (the product of *s* and *t*) at *x* = 6. In simpler terms, we substitute 6 into the product of the functions *s* and *t*.
Now, let’s consider the options provided and determine which one correctly represents the expression $(s \cdot t)(6)$.
Analyzing the Options
To determine the equivalent expression, let's analyze each option step by step:
A. $s(t(6))$
This option represents a composite function, not a product. Here, $t(6)$ means we first evaluate the function t at 6, yielding a value. Then, we use this value as the input for the function s. Symbolically, if $t(6) = a$, then $s(t(6))$ becomes $s(a)$. This is a completely different operation from multiplying the functions s and t.
To further illustrate this, consider an example. Let $s(x) = x + 1$ and $t(x) = 2x$. Then, $t(6) = 2 \cdot 6 = 12$, and $s(t(6)) = s(12) = 12 + 1 = 13$. This result comes from a composition, where the output of one function becomes the input of another. This is distinct from multiplying the functions' outputs at a specific point.
In summary, option A involves function composition, where the output of one function serves as the input for another, and it does not correctly represent the original expression, which involves pointwise multiplication of two functions evaluated at a specific point.
B. $s(x) \times t(6)$
This option presents a mix of function evaluation and a variable input. Here, we are evaluating function t at 6, resulting in a constant value, while function s remains a function of x. This expression implies that the output of function s for any given x is multiplied by the constant result of function t evaluated at 6.
To clarify, let’s use the same example functions as before: $s(x) = x + 1$ and $t(x) = 2x$. Then, $t(6) = 2 \cdot 6 = 12$, and $s(x) \times t(6)$ becomes $(x + 1) \times 12$, which simplifies to $12x + 12$. This expression is a function of x and will vary depending on the value of x. However, the original expression $(s \cdot t)(6)$ seeks a specific value resulting from both functions being evaluated at x = 6.
In essence, option B combines a general function evaluation, $s(x)$, with a specific evaluation of t at 6. This combination does not match the original expression where both functions should be evaluated at x = 6 before multiplication. Therefore, this option is not equivalent to $(s \cdot t)(6)$.
C. $s(6) \times t(6)$
This option precisely captures the meaning of $(s \cdot t)(6)$. It states that we should evaluate both functions s and t at x = 6, and then multiply the results. This aligns perfectly with the definition of pointwise multiplication of functions.
Let's revisit our example functions: $s(x) = x + 1$ and $t(x) = 2x$. Evaluating these at x = 6 gives us $s(6) = 6 + 1 = 7$ and $t(6) = 2 \cdot 6 = 12$. Multiplying these results, we get $s(6) \times t(6) = 7 \times 12 = 84$. This result is consistent with the original expression where we multiply the function outputs at the same point.
Thus, option C correctly represents the operation described in $(s \cdot t)(6)$, as it involves evaluating both functions at the same point and then multiplying the results. This is the very essence of pointwise multiplication of functions.
D. $6 \times s(x) \times t(x)$
This option presents a scalar multiplication combined with function multiplication. Here, we are multiplying the functions s and t as functions of x, and then multiplying the entire result by the scalar 6. This is different from evaluating the product of the functions at a specific point, which is what the original expression $(s \cdot t)(6)$ requires.
To illustrate, let's use our example functions: $s(x) = x + 1$ and $t(x) = 2x$. The expression $6 \times s(x) \times t(x)$ would translate to $6 \times (x + 1) \times (2x)$, which simplifies to $6 \times (2x^2 + 2x)$ or $12x^2 + 12x$. This is a quadratic function of x, representing a different kind of mathematical object compared to a single value that we would get from $(s \cdot t)(6)$.
Therefore, option D does not align with the original expression because it multiplies the functions generally as functions of x and then scales the result by 6, rather than evaluating the product of the functions at the specific point x = 6. Consequently, this option is not equivalent to $(s \cdot t)(6)$.
Conclusion
After analyzing each option, it is clear that option C, $s(6) \times t(6)$, is the expression equivalent to $(s \cdot t)(6)$. This is because it correctly represents the evaluation of both functions s and t at x = 6, followed by the multiplication of the results. The other options either involve function composition, a mix of function evaluation and variable input, or scalar multiplication combined with function multiplication, none of which accurately reflect the original expression.
Therefore, the correct answer is:
**C. $s(6) \times t(6)$
Which of the following expressions is equivalent to $(s )(6)$?
Equivalent Expressions for Function Products A Detailed Explanation