Question

homework 16: section 3.4
score: 107.5/160 answered: 11/16
question 13
given $f(x) = 5x + 5$ and $g(x) = 3x + 4$.
after simplifying,
a) $(f + g)(x) = $
b) $(f - g)(x) = $
c) $(fg)(x) = $
d) $(f \circ g)(x) = $
e) $(g \circ f)(x) =
question help: message instructor

Explanation:

Part (a)

Step1: Recall the definition of function addition

The sum of two functions \((f + g)(x)\) is defined as \(f(x)+g(x)\).
Given \(f(x)=5x + 5\) and \(g(x)=3x + 4\), we substitute these into the formula:
\((f + g)(x)=f(x)+g(x)=(5x + 5)+(3x + 4)\)

Step2: Combine like terms

Combine the \(x\)-terms and the constant terms:
\(5x+3x = 8x\) and \(5 + 4=9\)
So, \((f + g)(x)=8x + 9\)

Part (b)

Step1: Recall the definition of function subtraction

The difference of two functions \((f - g)(x)\) is defined as \(f(x)-g(x)\).
Substitute \(f(x)=5x + 5\) and \(g(x)=3x + 4\) into the formula:
\((f - g)(x)=f(x)-g(x)=(5x + 5)-(3x + 4)\)

Step2: Distribute the negative sign and combine like terms

Distribute the negative sign: \(5x+5 - 3x-4\)
Combine like terms: \(5x-3x=2x\) and \(5 - 4 = 1\)
So, \((f - g)(x)=2x+1\)

Part (c)

Step1: Recall the definition of function multiplication

The product of two functions \((fg)(x)\) is defined as \(f(x)\times g(x)\).
Substitute \(f(x)=5x + 5\) and \(g(x)=3x + 4\) into the formula:
\((fg)(x)=(5x + 5)(3x + 4)\)

Step2: Use the distributive property (FOIL method)

First: \(5x\times3x = 15x^{2}\)
Outer: \(5x\times4=20x\)
Inner: \(5\times3x = 15x\)
Last: \(5\times4 = 20\)
Combine like terms: \(20x+15x = 35x\)
So, \((fg)(x)=15x^{2}+35x + 20\)

Part (d)

Answer:

s:
a) \(\boldsymbol{8x + 9}\)
b) \(\boldsymbol{2x + 1}\)
c) \(\boldsymbol{15x^{2}+35x + 20}\)
d) \(\boldsymbol{15x + 25}\)
e) \(\boldsymbol{15x + 19}\)