Let f(x) = m1x + b1 and g(x) = m2x + b2 be two linear functions.
Answer the following, noting that answers may depend on the the ms and bs.
a. Is f (x) + g(x) a linear function of x?
b. Is f (x) − g(x) a linear function of x?
c. Is f (x) × g(x) a linear function of x?
d. Is f(x)/g(x) a linear function of x?
Why don't you try it with two different functions,
e.g.
f(x) = x+2 , g(x) = 2x - 1
let y = f(x) + g(x)
= x+2 + 2x - 1
= 3x + 1
is this a straight line ?
so what do you think about a) ???
...
c) let y = f(x) * g(x)
= (x+2)(2x-1) = 2x^2 + 3x - 2
Is a quadratic a straight line, that is, a linear function??
etc
a. No, f(x) + g(x) is not a linear function of x. When you add two linear functions, the resulting function may have a non-zero constant term, which means it is not of the form mx + b and therefore not linear.
b. Yes, f(x) - g(x) is a linear function of x. When you subtract two linear functions, the resulting function will still be of the form mx + b, so it remains linear.
c. No, f(x) × g(x) is not a linear function of x. When you multiply two linear functions, the resulting function will have a quadratic term (x^2) as well as higher order terms, which makes it nonlinear.
d. No, f(x)/g(x) is not a linear function of x. When you divide two linear functions, the resulting function will have a rational expression instead of a linear equation, so it is not linear.
To answer each of these questions, we need to determine if the given expression results in a linear function of x.
a. To find out if f(x) + g(x) is a linear function of x, we need to check if the sum of the two functions can be written in the form mx + b, where m and b are constants.
To find f(x) + g(x), we simply add the two functions:
f(x) + g(x) = (m1x + b1) + (m2x + b2)
Combining like terms, we get:
= (m1 + m2)x + (b1 + b2)
Since the resulting expression can be written in the form mx + b, we conclude that f(x) + g(x) is a linear function of x.
b. Similarly, to determine if f(x) - g(x) is a linear function of x, we subtract the two functions:
f(x) - g(x) = (m1x + b1) - (m2x + b2)
Combining like terms, we get:
= (m1 - m2)x + (b1 - b2)
Since the resulting expression can be written in the form mx + b, we conclude that f(x) - g(x) is a linear function of x.
c. For f(x) * g(x) to be a linear function of x, we would need the product of the two functions to be in the form mx + b. However, multiplying two linear functions will result in a quadratic function, not a linear one. Therefore, we can conclude that f(x) * g(x) is not a linear function of x.
d. Dividing f(x) by g(x), f(x)/g(x), also does not result in a linear function. Division of linear functions leads to a fractional expression, which generally cannot be represented in the form mx + b. Hence, f(x)/g(x) is not a linear function of x.