Question

Suppose that the functions s and t are defined for all real numbers x as follows.

s(x) = x + 5
t(x) = 2x + 6

Write the expressions for (s ⋅ t)(x) and (s + t)(x) and evaluate (s ⋅ t)(-2) .

a) (s ⋅ t)(x) = 2x^2 + 16x + 30 , (s + t)(x) = 3x + 11 , (s ⋅ t)(-2) = -6
b) (s ⋅ t)(x) = 2x^2 + 16x + 30 , (s + t)(x) = 3x + 11 , (s ⋅ t)(-2) = -10
c) (s ⋅ t)(x) = x^2 + 11x + 30 , (s + t)(x) = 3x + 11 , (s ⋅ t)(-2) = -6
d) (s ⋅ t)(x) = x^2 + 11x + 30 , (s + t)(x) = 3x + 11 , (s ⋅ t)(-2) = -10

Answer 1

The derived functions from s(x) = x + 5 and t(x) = 2x + 6 are (s ⋅ t)(x) = 2x^2 + 16x + 30 and (s + t)(x) = 3x + 11. Evaluating (s ⋅ t)(-2) yields a result of 6. However, the provided options in the question do not list this result, indicating a possible typo.

Step-by-step explanation:

The question asks us to find expressions for two functions derived from the original functions s(x) = x + 5 and t(x) = 2x + 6. The first derived function is the product of s and t, denoted as (s ⋅ t)(x). The second derived function is the sum of s and t, denoted as (s + t)(x). Additionally, we are asked to evaluate (s ⋅ t)(-2).

To find (s ⋅ t)(x), we multiply the functions s(x) and t(x) together:

• 
  
• s(x) = x + 5
• 
  
• t(x) = 2x + 6

(s ⋅ t)(x) = (x + 5)(2x + 6) = 2x2 + 16x + 30

To find (s + t)(x), we add the functions together:

(s + t)(x) = (x + 5) + (2x + 6) = 3x + 11

Now, to evaluate (s ⋅ t)(-2), we substitute -2 into our found product:

(s ⋅ t)(-2) = 2(-2)2 + 16(-2) + 30 = 8 - 32 + 30 = 6

However, looking at the possible choices, we can see that none of the options indicate (s ⋅ t)(-2) = 6. Therefore, it seems there may be a typo, as the correct evaluation is not listed among the provided options.