Question

suppose that the functions $s$ and $t$ are defined for all real numbers $x$ as follows.$s(x)=x^2$$t(x)=3x^3$write the expressions for $(s-t)(x)$ and $(s\cdot t)(x)$ and evaluate $(s+t)(-2)$.$(s-t)(x)$$(s\cdot t)(x)$$(s+t)(-2)$

Explanation:

Step1: Define $(s-t)(x)$

$(s-t)(x) = s(x) - t(x)$

Step2: Substitute function expressions

$(s-t)(x) = x^2 - 3x^3$

Step3: Define $(s \cdot t)(x)$

$(s \cdot t)(x) = s(x) \cdot t(x)$

Step4: Multiply the functions

$(s \cdot t)(x) = x^2 \cdot 3x^3 = 3x^{2+3} = 3x^5$

Step5: Define $(s+t)(-2)$

$(s+t)(-2) = s(-2) + t(-2)$

Step6: Compute $s(-2)$ and $t(-2)$

$s(-2) = (-2)^2 = 4$, $t(-2) = 3(-2)^3 = 3(-8) = -24$

Step7: Sum the results

$(s+t)(-2) = 4 + (-24) = -20$

Answer:

$(s-t)(x) = x^2 - 3x^3$
$(s \cdot t)(x) = 3x^5$
$(s+t)(-2) = -20$