Math Problem Statement

The blue and black curves on the graph below represent f(x) and g(x), respectively. Find (f ∘ g)(-4).

Solution

The problem asks to find (fg)(4)(f \circ g)(-4), which is the composition of the functions f(x)f(x) and g(x)g(x), evaluated at x=4x = -4. Here's how to solve this:

  1. Interpret the composition: (fg)(4)=f(g(4))(f \circ g)(-4) = f(g(-4)) This means we first need to find g(4)g(-4), and then substitute that result into the function ff.

  2. Find g(4)g(-4): From the graph, g(x)g(x) is represented by the black curve. When x=4x = -4, locate the corresponding yy-value on the black curve.

    By observing the black curve at x=4x = -4, the yy-coordinate appears to be approximately g(4)=2g(-4) = 2.

  3. Find f(g(4))=f(2)f(g(-4)) = f(2): Now, substitute g(4)=2g(-4) = 2 into f(x)f(x). The function f(x)f(x) is represented by the blue curve. Look for the yy-value when x=2x = 2 on the blue curve.

    From the graph, when x=2x = 2 on the blue curve, the yy-value appears to be approximately f(2)=4f(2) = 4.

Thus, (fg)(4)=f(2)=4(f \circ g)(-4) = f(2) = 4.

Would you like any more details or have questions on this process?

Here are 5 related questions for further practice:

  1. What is (gf)(2)(g \circ f)(2) using the same graph?
  2. Can you find (fg)(6)(f \circ g)(-6)?
  3. What is the value of g(f(2))g(f(2))?
  4. At what xx-value does f(x)=0f(x) = 0?
  5. For what value of xx is g(x)=2g(x) = -2?

Tip: When evaluating a composite function from a graph, it's important to track the input-output flow of values carefully, first through one function and then the next.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Graph Interpretation

Formulas

(f ∘ g)(x) = f(g(x))

Theorems

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Suitable Grade Level

Grades 9-12