Question

The function f is defined as follows: f (x)= 4x^2-8x-2

Express f(x) in the form a(x − h)^2 + k where a, h and k are constants

Answer 1

Answer:

Multiplying two functions

Example

Let's look an example to see how this works.

Given that f(x)=2x-3f(x)=2x−3f, left parenthesis, x, right parenthesis, equals, 2, x, minus, 3 and g(x)=x+1g(x)=x+1g, left parenthesis, x, right parenthesis, equals, x, plus, 1, find (f\cdot g)(x)(f⋅g)(x)left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis.

Solution

The most difficult part of combining functions is understanding the notation. What does (f\cdot g)(x)(f⋅g)(x)left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis mean?

Well, (f\cdot g)(x)(f⋅g)(x)left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis just means to find the product of f(x)f(x)f, left parenthesis, x, right parenthesis and g(x)g(x)g, left parenthesis, x, right parenthesis. Mathematically, this means that (f\cdot g)(x)=f(x)\cdot g(x)(f⋅g)(x)=f(x)⋅g(x)left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis, dot, g, left parenthesis, x, right parenthesis.

Now, this becomes a familiar problem.

\begin{aligned} (f\cdot g)(x) &= f(x)\cdot g(x)&\gray{\text{Define.}} \\\\ &= \left(2x-3\right)\cdot\left(x+1\right) &\gray{\text{Substitute.}} \\\\ &= 2x^2+2x-3x-3&\gray{\text{Distribute.}} \\\\ &=2x^2-x-3&\gray{\text{Combine like terms.}} \end{aligned}  

(f⋅g)(x)

​

 

=f(x)⋅g(x)

=(2x−3)⋅(x+1)

=2x  

2

+2x−3x−3

=2x  

2

−x−3

​

 

Define.

Substitute.

Distribute.

Combine like terms.

​

 

Note: We simplified the result to obtain a nicer expression, but this is not necessary.

Step-by-step explanation:

Answer 2

Answer:

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