Let's evaluate the functions $f(x)$ and $g(x)$ based on the provided graphs for the exercises 51-54.

From the graphs:

• The graph of $f(x)$ is a "V" shape:

  • $f(1) = 2$
  • $f(2) = 0$
  • $f(3) = 2$
  • $f(4) = 4$

• The graph of $g(x)$ is a straight line with a negative slope:

  • $g(1) = 4$
  • $g(2) = 3$
  • $g(3) = 2$
  • $g(4) = 1$

Let's evaluate each question:

51. (a) $(f + g)(3)$

$(f + g)(3) = f(3) + g(3) = 2 + 2 = 4$

(b) $\left(\frac{f}{g}\right)(2)$ $\left(\frac{f}{g}\right)(2) = \frac{f(2)}{g(2)} = \frac{0}{3} = 0$

52. (a) $(f - g)(1)$

$(f - g)(1) = f(1) - g(1) = 2 - 4 = -2$

(b) $(fg)(4)$ $(fg)(4) = f(4) \cdot g(4) = 4 \cdot 1 = 4$

53. (a) $(f \circ g)(2)$

$(f \circ g)(2) = f(g(2)) = f(3) = 2$

(b) $(g \circ f)(2)$ $(g \circ f)(2) = g(f(2)) = g(0)$ $g(0)$ is not defined on the graph, so this is undefined.

54. (a) $(f \circ g)(1)$

$(f \circ g)(1) = f(g(1)) = f(4) = 4$

(b) $(g \circ f)(3)$ $(g \circ f)(3) = g(f(3)) = g(2) = 3$

Would you like further details on any of these computations?

Related Questions:

1. What is the domain and range of the functions $f(x)$ and $g(x)$?
2. How would the composition $(f \circ g)(x)$ be generally evaluated for any $x$?
3. Can $g \circ f(x)$ have undefined values, and why?
4. What happens when you add, subtract, or multiply two functions graphically?
5. How can you use these graphs to approximate the derivative of $f(x)$ or $g(x)$?

Tip:

When composing two functions, always evaluate the inner function first and ensure that the result falls within the domain of the outer function.