Answer:
Option (a) is correct.
The smallest coefficient on x - terms will be obtained by addition and multiplication of two given functions.
Step-by-step explanation:
Given: The function \(f(x)=-2x-1\) and \(g(x)=-3x+4\)
We have to find the operation from the given options that results in the smallest coefficient on the x term .
Consider the given function \(f(x)=-2x-1\) and \(g(x)=-3x+4\)
b) Subtraction
That is f(x) - g(x)
\(f(x) -g(x)=-2x-1-(-3x+4)\)
Apply plus - minus rule -(-a) = a , we have,
\(f(x)-g(x)=-2x-1+3x-4\)
\(f(x)+g(x)=x-5\)
Here, The coefficient of x is 1.
c) Multiplication
That is \(f(x)\cdot g(x)=(-2x-1)\cdot (-3x+4)\)
\(\mathrm{Apply\:FOIL\:method}:\quad \left(a+b\right)\left(c+d\right)=ac+ad+bc+bd\)
\(=\left(-2x\right)\left(-3x\right)+\left(-2x\right)\cdot \:4+\left(-1\right)\left(-3x\right)+\left(-1\right)\cdot \:4\)
Simplify, we have,
\(=6x^2-5x-4\)
Here, The coefficient of x is -5.
d) Addition
That is f(x) + g(x)
\(f(x)+g(x)=-2x-1+(-3x+4)\)
Simplify, we have,
\(f(x)+g(x)=-2x-1-3x+4\)
\(f(x)+g(x)=-5x+3\)
Here, The coefficient of x is -5.
a) two operations result in the same coefficient.
When we add or multiply the two given function, we obtain the same coefficient of x that is -5
Hence, The smallest coefficient on x - terms will be obtained by addition and multiplication of two given functions.