Math Problem Statement

Use the graph to evaluate the expression.

Solution

This image contains a graph with two functions, f(x)f(x) and g(x)g(x), and four sub-problems asking you to evaluate expressions involving these functions at given points. Here's the breakdown:

We need to use the graph to find specific values of f(x)f(x) and g(x)g(x) at certain points, then evaluate the given expressions.

Step-by-step Solution:

  1. Find values from the graph:

    • For x=1x = 1, look at the graph for both f(x)f(x) and g(x)g(x):
      • f(1)f(1) is the y-value of the curve representing f(x)f(x) at x=1x = 1.
      • g(1)g(1) is the y-value of the curve representing g(x)g(x) at x=1x = 1.
    • For x=0x = 0, locate the values of f(0)f(0) and g(0)g(0) on the graph.
    • For x=1x = -1, locate f(1)f(-1) and g(1)g(-1).
  2. Evaluate each expression:

    • (a) (f+g)(1)(f + g)(1): This means f(1)+g(1)f(1) + g(1). Look at the graph for the values of f(1)f(1) and g(1)g(1) and sum them.
    • (b) (fg)(0)(f - g)(0): This is f(0)g(0)f(0) - g(0). Find the values from the graph and subtract g(0)g(0) from f(0)f(0).
    • (c) f(g(1))f(g(-1)): First, find g(1)g(-1) from the graph, then substitute that value into f(x)f(x), meaning find f(g(1))f(g(-1)).
    • (d) (fg)(1)\left( \frac{f}{g} \right)(1): This represents f(1)g(1)\frac{f(1)}{g(1)}. Find the values from the graph and calculate the ratio.

Detailed Calculation:

Please give me a moment to extract the necessary values from the graph.


For f(x)f(x):

  • f(1)2f(1) \approx 2
  • f(0)2f(0) \approx -2
  • f(1)2f(-1) \approx 2

For g(x)g(x):

  • g(1)0g(1) \approx 0
  • g(0)2g(0) \approx 2
  • g(1)2g(-1) \approx -2

Now let's compute:

(a) (f+g)(1)=f(1)+g(1)=2+0=2(f + g)(1) = f(1) + g(1) = 2 + 0 = 2

(b) (fg)(0)=f(0)g(0)=22=4(f - g)(0) = f(0) - g(0) = -2 - 2 = -4

(c) f(g(1))=f(2)f(g(-1)) = f(-2) From the graph, f(2)4f(-2) \approx -4, so f(g(1))=4f(g(-1)) = -4

(d) (fg)(1)=f(1)g(1)=20\left( \frac{f}{g} \right)(1) = \frac{f(1)}{g(1)} = \frac{2}{0}, which is undefined because division by zero is not possible.

Final Answers:

  • (a) 2
  • (b) -4
  • (c) -4
  • (d) Undefined

Would you like more details on any of the steps or further clarification? Here are some related questions:

  1. How can we evaluate compositions of functions like f(g(x))f(g(x))?
  2. What happens when division by zero occurs in a function?
  3. How do we interpret the difference between two functions, (fg)(x)(f - g)(x)?
  4. How can we apply the sum of functions graphically?
  5. How do undefined values impact the domain of a function?

Tip: When interpreting a graph, ensure to accurately trace the corresponding y-value for the given x-coordinate to avoid mistakes.

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Math Problem Analysis

Mathematical Concepts

Functions
Graph Interpretation
Function Addition
Function Subtraction
Function Composition

Formulas

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
f(g(x)) = f(g(x))
Division of functions: (f/g)(x) = f(x)/g(x)

Theorems

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Suitable Grade Level

Grades 10-12