Question

1. find the value of b given f(x)=3x² - bx + 4 and f(-1)=9. 5. state whether g(x)=-2|x|+3x² represents an even function, an odd function or neither. your method must show an understanding of the algebraic rule.

Explanation:

4.

Step1: Substitute x = - 1 into f(x)

Given \(f(x)=3x^{2}-bx + 4\) and \(f(-1)=9\). Substitute \(x=-1\) into \(f(x)\): \(f(-1)=3\times(-1)^{2}-b\times(-1)+4\).

Step2: Simplify the equation

\[

$$\begin{align*} 3\times(-1)^{2}-b\times(-1)+4&=9\\ 3\times1 + b+4&=9\\ 3 + b+4&=9\\ b + 7&=9 \end{align*}$$

\]

Step3: Solve for b

Subtract 7 from both sides of the equation \(b + 7=9\): \(b=9 - 7\).

Step1: Recall the definitions of even and odd functions

An even function satisfies \(g(-x)=g(x)\) for all \(x\) in the domain, and an odd function satisfies \(g(-x)=-g(x)\) for all \(x\) in the domain. Given \(g(x)=-2|x|+3x^{2}\), find \(g(-x)\).

Step2: Calculate g(-x)

Substitute \(-x\) into \(g(x)\): \(g(-x)=-2|-x|+3(-x)^{2}\). Since \(|-x| = |x|\) and \((-x)^{2}=x^{2}\), we have \(g(-x)=-2|x|+3x^{2}\).

Step3: Compare g(-x) with g(x)

Since \(g(-x)=-2|x|+3x^{2}=g(x)\), \(g(x)\) is an even function.

Answer:

\(b = 2\)

5.