Sum, difference, and product of two functions Suppose that the functions \( r \) and \( s \) are defined for all real numbers \( x \) as follows. \[ \begin{array}{r}r(x)=3 x^{2} \\ s(x)=2 x\end{array} \] Write the expressions for \( (s \cdot r)(x) \) and \( (s+r)(x) \) and evaluate \( (s-r)(-1) \).

To find the expressions for the sum, difference, and product of the functions \( r \) and \( s \), we start with the definitions of the functions: \[ r(x) = 3x^2 \] \[ s(x) = 2x \] ### Sum of the Functions The sum of the functions \( (s + r)(x) \) is given by: \[ (s + r)(x) = s(x) + r(x) = 2x + 3x^2 \] ### Difference of the Functions The difference of the functions \( (s - r)(x) \) is given by: \[ (s - r)(x) = s(x) - r(x) = 2x - 3x^2 \] ### Product of the Functions The product of the functions \( (s \cdot r)(x) \) is given by: \[ (s \cdot r)(x) = s(x) \cdot r(x) = (2x)(3x^2) = 6x^3 \] ### Evaluating the Difference at \( x = -1 \) Now, we need to evaluate \( (s - r)(-1) \): \[ (s - r)(-1) = 2(-1) - 3(-1)^2 \] Calculating each term: 1. \( 2(-1) = -2 \) 2. \( 3(-1)^2 = 3(1) = 3 \) Now substituting these values back into the expression: \[ (s - r)(-1) = -2 - 3 = -5 \] ### Summary of Results - \( (s + r)(x) = 2x + 3x^2 \) - \( (s - r)(x) = 2x - 3x^2 \) - \( (s \cdot r)(x) = 6x^3 \) - \( (s - r)(-1) = -5 \)