If [tex]s(x)=x-7[/tex] and [tex]t(x)=4 x^2-x+3[/tex], which expression is equivalent to [tex](t \circ s)(x)[/tex]?

A. [tex]4(x-7)^2-x-7+3[/tex]
B. [tex]4(x-7)^2-(x-7)+3[/tex]
C. [tex](4 x^2-x+3)-7[/tex]
D. [tex](4 x^2-x+3)(x-7)[/tex]

Find s ( x ) : s ( x ) = x − 7 .
Substitute s ( x ) into t ( x ) : t ( s ( x )) = 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .
Compare the result with the given options.
The equivalent expression is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 ​ .

Explanation

Understanding the Problem We are given two functions, s ( x ) = x − 7 and t ( x ) = 4 x 2 − x + 3 . We need to find the expression equivalent to ( t ∘ s ) ( x ) , which means t ( s ( x )) .

Finding s(x) First, we find s ( x ) , which is given as s ( x ) = x − 7 .

Substituting s(x) into t(x) Next, we substitute s ( x ) into t ( x ) to find t ( s ( x )) . This means we replace every x in t ( x ) with ( x − 7 ) . So, we have


t ( s ( x )) = t ( x − 7 ) = 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .

Comparing with the Options Now, we compare our result with the given options:

4 ( x − 7 ) 2 − x − 7 + 3 4 ( x − 7 ) 2 − ( x − 7 ) + 3 ( 4 x 2 − x + 3 ) − 7 ( 4 x 2 − x + 3 ) ( x − 7 )
Our expression 4 ( x − 7 ) 2 − ( x − 7 ) + 3 matches the second option.

Final Answer Therefore, the expression equivalent to ( t ∘ s ) ( x ) is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 .

Examples
Composition of functions is a fundamental concept in mathematics and has many real-world applications. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 8%. If x is the original price, the discounted price is s ( x ) = 0.9 x , and the price after tax is t ( x ) = 1.08 x . The final price after both discount and tax is ( t ∘ s ) ( x ) = t ( s ( x )) = 1.08 ( 0.9 x ) = 0.972 x . This shows how composition of functions can model sequential operations.

Answered by GinnyAnswer | 2025-07-03

The expression equivalent to ( t ∘ s ) ( x ) is 4 ( x − 7 ) 2 − ( x − 7 ) + 3 , matching option B. We found this by substituting s ( x ) into t ( x ) and simplifying the resulting expression. Thus, the answer is B ​ .
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Answered by Anonymous | 2025-07-04