Math Problem Statement

Given f(x) = -x + 1 and g(x) = 4x + 2, calculate the value of (-f(x) + 2g(x)) at x = 3.

Solution

Let’s analyze the problem in the image and solve it step by step.


We are given two functions:

  1. f(x)=x+1f(x) = -x + 1
  2. g(x)=4x+2g(x) = 4x + 2

The task is to calculate the value of:
(f(x)+2g(x))x=3(-f(x) + 2g(x)) \big|_{x=3}


Step 1: Break Down the Expression

The expression to evaluate is: f(x)+2g(x)-f(x) + 2g(x)

This means: f(x)=(x+1)=x1-f(x) = -(-x + 1) = x - 1
and
2g(x)=2(4x+2)=8x+42g(x) = 2(4x + 2) = 8x + 4

Thus: f(x)+2g(x)=(x1)+(8x+4)=9x+3-f(x) + 2g(x) = (x - 1) + (8x + 4) = 9x + 3


Step 2: Evaluate at x=3x = 3

Now, substitute x=3x = 3 into 9x+39x + 3: 9(3)+3=27+3=309(3) + 3 = 27 + 3 = 30


Final Answer:

The value of (f(x)+2g(x))x=3(-f(x) + 2g(x)) \big|_{x=3} is: 30\boxed{30}


Would you like further details on any step? Here are 5 related questions you could explore:

  1. How do you find the derivative of f(x)f(x) or g(x)g(x)?
  2. What happens if x=3x = -3? Can the same method apply?
  3. Can this problem be generalized for any xx?
  4. What is the geometric interpretation of f(x)f(x) and g(x)g(x)?
  5. How does f(x)+2g(x)-f(x) + 2g(x) behave as xx \to \infty?

Tip: Always simplify the given expression before substituting values. It makes calculations more manageable.

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Math Problem Analysis

Mathematical Concepts

Functions
Linear Algebra
Substitution

Formulas

f(x) = -x + 1
g(x) = 4x + 2
Simplification and evaluation of expressions

Theorems

Basic arithmetic rules
Function evaluation

Suitable Grade Level

Grades 8-10