Question

f(x) = -5x
g(x) = 8x² - 5x - 9
find (f · g)(x).
-40x³ + 25x² + 45x
-40x² + 25x + 45x
-40x⁴ + 25x³ + 45x²
-40x³ - 5x - 9

Explanation:

Step1: Recall the product rule for functions

To find \((f \cdot g)(x)\), we use the formula \((f \cdot g)(x)=f(x)\times g(x)\). Given \(f(x) = - 5x\) and \(g(x)=8x^{2}-5x - 9\), we need to multiply \(-5x\) with each term in \(g(x)\).

Step2: Multiply \(-5x\) by each term of \(g(x)\)

• Multiply \(-5x\) with \(8x^{2}\): \(-5x\times8x^{2}=-40x^{3}\) (using the rule \(a^{m}\times a^{n}=a^{m + n}\), here \(x\times x^{2}=x^{1+2}=x^{3}\))
• Multiply \(-5x\) with \(-5x\): \(-5x\times(-5x) = 25x^{2}\) (since \((-5)\times(-5) = 25\) and \(x\times x=x^{2}\))
• Multiply \(-5x\) with \(-9\): \(-5x\times(-9)=45x\) (since \((-5)\times(-9) = 45\))

Step3: Combine the results

\((f\cdot g)(x)=-40x^{3}+25x^{2}+45x\)

Answer:

\(-40x^{3}+25x^{2}+45x\) (corresponding to the first option among the given choices with this expression)