Question

Let f and g be linear functions with equations f(x) = m1x + b1 and g(x) = m2x + b2.

(a) Is f of g also a linear function?
b) if so what is the slope

Answer 1

Yes, the composition of two linear functions is also a linear function. The slope of the composed function is the product of the slopes of the individual functions, therefore the slope of f(g(x)) is m1*m2.

Step-by-step explanation:

When composing two linear functions, such as f(x) = m1x + b1 and g(x) = m2x + b2, the result is another linear function. This is because the composition of the two functions, f(g(x)), involves inserting one linear function into the other, leading to operations that are characteristic of linear functions: addition, subtraction, and multiplication by constants.

To find the slope of f(g(x)), we substitute g(x) into f(x), resulting in f(g(x)) = m1(m2x + b2) + b1. After applying the distributive property, we get f(g(x)) = (m1*m2)x + (m1*b2 + b1). Therefore, the slope of the composed function is m1*m2.

Answer 2

f(x) = m1x + b1
g(x) = m2x + b2

f(g(x)) = f(m2x + b2) = m1(m2x + b2) + b1 = m1m2x + b2 + b1

a.) f of g is also a linear function.

b.) slope is m1m2