Question

pre-calculus
chapter 1 team test
name:

1. with rising energy prices, the (gloggum’s?) are worried about how much it costs each time they heat up their hot tub. they keep it unheated unless they want to use it. they kept track of how many days they used the hot tub each month, along with their energy bill. using the data they collected below, find a linear model for the data and determine approximately how much it costs each time the (gloggum’s?) use their hot tub.

days used | energy bill

15 | $235
13 | $229
14 | $230
11 | $220

days used | energy bill

8 | $213
6 | $215
2 | $201
2 | $205

days used | energy bill

5 | $211
7 | $218
12 | $223
17 | $241

1. given ( f(x) = x^2 + 3x ) and ( g(x) = x - 5 ), find all values of ( x ) such that ( f(g(x)) = f(x) + g(x) ). show all of your steps!

handwritten: ( f(g(x-5)) = (x^2 + 3x) + (x - 5) ), ( = x^3 + 3x^2 - 5x^2 -15x ), ( = x^3 - 2x^2 -15x )

Explanation:

Problem 2 Solution:

Step 1: Find \( f(g(x)) \)

First, substitute \( g(x) = x - 5 \) into \( f(x) \). So \( f(g(x)) = f(x - 5) \).
Since \( f(t) = t^2 + 3t \) (let \( t = x - 5 \)), we have:
\( f(x - 5) = (x - 5)^2 + 3(x - 5) \)
Expand \( (x - 5)^2 \): \( x^2 - 10x + 25 \)
Expand \( 3(x - 5) \): \( 3x - 15 \)
Combine terms: \( x^2 - 10x + 25 + 3x - 15 = x^2 - 7x + 10 \)

Step 2: Find \( f(x) + g(x) \)

Given \( f(x) = x^2 + 3x \) and \( g(x) = x - 5 \), add them:
\( f(x) + g(x) = (x^2 + 3x) + (x - 5) = x^2 + 4x - 5 \)

Step 3: Set \( f(g(x)) = f(x) + g(x) \) and solve for \( x \)

Set \( x^2 - 7x + 10 = x^2 + 4x - 5 \)
Subtract \( x^2 \) from both sides: \( -7x + 10 = 4x - 5 \)
Add \( 7x \) to both sides: \( 10 = 11x - 5 \)
Add 5 to both sides: \( 15 = 11x \)
Divide by 11: \( x = \frac{15}{11} \)

Answer:

\( x = \frac{15}{11} \) (or approximately \( 1.36 \))