Math Problem Statement

Complete the table of values for the functions f(x)=|x–3|–4 and g(x)=3x–5. Write your answers as whole numbers, decimals, or simplified fractions or mixed numbers. x f(x) g(x) 0 – 1 – 4 1 2 – 3 4 3 4 Based on the values in the table, where does the equation f(x)=g(x) have a solution?

Solution

Let's complete the table step by step by evaluating each function at the given values of xx.

Given functions:

f(x)=x34f(x) = |x - 3| - 4 g(x)=3x5g(x) = 3x - 5

Step 1: Compute f(x)f(x) and g(x)g(x) for each xx.

For x=0x = 0: f(0)=034=34=34=1f(0) = |0 - 3| - 4 = | -3 | - 4 = 3 - 4 = -1 g(0)=3(0)5=05=5g(0) = 3(0) - 5 = 0 - 5 = -5

For x=1x = 1: f(1)=134=24=24=2f(1) = |1 - 3| - 4 = | -2 | - 4 = 2 - 4 = -2 g(1)=3(1)5=35=2g(1) = 3(1) - 5 = 3 - 5 = -2

For x=2x = 2: f(2)=234=14=14=3f(2) = |2 - 3| - 4 = | -1 | - 4 = 1 - 4 = -3 g(2)=3(2)5=65=1g(2) = 3(2) - 5 = 6 - 5 = 1

For x=3x = 3: f(3)=334=04=04=4f(3) = |3 - 3| - 4 = |0| - 4 = 0 - 4 = -4 g(3)=3(3)5=95=4g(3) = 3(3) - 5 = 9 - 5 = 4

For x=4x = 4: f(4)=434=14=14=3f(4) = |4 - 3| - 4 = |1| - 4 = 1 - 4 = -3 g(4)=3(4)5=125=7g(4) = 3(4) - 5 = 12 - 5 = 7

Completed Table:

xxf(x)f(x)g(x)g(x)
0-1-5
1-2-2
2-31
3-44
4-37

Step 2: Solve f(x)=g(x)f(x) = g(x)

We find that f(x)=g(x)f(x) = g(x) when x=1x = 1, since: f(1)=2,g(1)=2f(1) = -2, \quad g(1) = -2

Thus, the equation f(x)=g(x)f(x) = g(x) has a solution at x=1x = 1.


Would you like a graphical explanation of this solution? Here are some related questions:

  1. How do you determine where two functions are equal algebraically?
  2. What is the geometric interpretation of f(x)=g(x)f(x) = g(x)?
  3. Can we solve f(x)=g(x)f(x) = g(x) without making a table?
  4. How does the absolute value function affect the behavior of f(x)f(x)?
  5. What happens if we extend the table to more values of xx?

Tip: When comparing two functions, their intersection points give the solutions to f(x)=g(x)f(x) = g(x).

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Math Problem Analysis

Mathematical Concepts

Absolute Value Functions
Linear Functions
Solving Equations

Formulas

f(x) = |x - 3| - 4
g(x) = 3x - 5

Theorems

Equality of functions

Suitable Grade Level

Grades 7-9