Math Problem Statement

Based on the values in the table, where does the equation f(x) = g(x) have a solution?

Solution

From the table provided in the image, we are tasked to determine where f(x)=g(x)f(x) = g(x) has a solution, given:

  • f(x)=x+5f(x) = |x| + 5
  • g(x)=(12)xg(x) = \left(\frac{1}{2}\right)^x

Table of values:

xxf(x)f(x)g(x)g(x)
-388
-274
-162
051
1612\frac{1}{2}

Step-by-Step Solution:

  1. Where f(x)=g(x)f(x) = g(x):

    • At x=3x = -3: f(x)=8f(x) = 8 and g(x)=8g(x) = 8, so they are equal at x=3x = -3.
  2. Between intervals:

    • For x=2x = -2: f(x)=7f(x) = 7 and g(x)=4g(x) = 4. Since f(x)>g(x)f(x) > g(x), there is no equality here.
    • For x=1x = -1: f(x)=6f(x) = 6 and g(x)=2g(x) = 2. Again, f(x)>g(x)f(x) > g(x).
    • For x=0x = 0: f(x)=5f(x) = 5 and g(x)=1g(x) = 1. Still, f(x)>g(x)f(x) > g(x).
  3. Conclusion:

    • f(x)=g(x)f(x) = g(x) at x=3x = -3.
    • The correct answer is x=3x = -3.

Do you need further clarification or a deeper breakdown of the calculations?

Related Questions:

  1. What happens to f(x)=x+5f(x) = |x| + 5 as xx \to \infty?
  2. What happens to g(x)=(12)xg(x) = \left(\frac{1}{2}\right)^x as xx \to -\infty?
  3. How can you solve f(x)=g(x)f(x) = g(x) algebraically instead of using a table?
  4. How does g(x)g(x) compare to f(x)f(x) for x>1x > 1?
  5. Can f(x)f(x) and g(x)g(x) intersect more than once for xRx \in \mathbb{R}?

Tip:

Always analyze the behavior of the functions at extreme values (like xx \to \infty or xx \to -\infty) to understand their trends!

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Math Problem Analysis

Mathematical Concepts

Algebra
Piecewise Functions
Exponential Functions

Formulas

f(x) = |x| + 5
g(x) = (1/2)^x

Theorems

Equality of Functions
Exponential Decay Behavior

Suitable Grade Level

Grades 8-10