Question

evaluate each expression using the graphs of \\(y = f(x)\\) and \\(y = g(x)\\) shown.

(a) \\((g \circ f)(-1)\\)
(b) \\((g \circ f)(0)\\)
(c) \\((f \circ g)(-1)\\)
(d) \\((f \circ g)(4)\\)

Explanation:

Evaluate (g ∘ f)(-1)

Using the Composite Functions knowledge point

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

From the graph of \(y = f(x)\), the point at \(x = -1\) is \((-1, 1)\), so:

$$ f(-1) = 1 $$

Now evaluate \(g(1)\). From the graph of \(y = g(x)\), the point at \(x = 1\) is \((1, 2)\), so:

$$ g(1) = 2 $$

Thus:

$$ (g \circ f)(-1) = 2 $$

Evaluate (g ∘ f)(0)

Using the Composite Functions knowledge point

$$ (g \circ f)(0) = g(f(0)) $$

From the graph of \(y = f(x)\), the point at \(x = 0\) is \((0, 0)\), so:

$$ f(0) = 0 $$

Now evaluate \(g(0)\). From the graph of \(y = g(x)\), the point at \(x = 0\) is \((0, 4)\), so:

$$ g(0) = 4 $$

Thus:

$$ (g \circ f)(0) = 4 $$

Evaluate (f ∘ g)(-1)

Using the Composite Functions knowledge point

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

From the graph of \(y = g(x)\), the point at \(x = -1\) is \((-1, 2)\), so:

$$ g(-1) = 2 $$

Now evaluate \(f(2)\). From the graph of \(y = f(x)\), the point at \(x = 2\) is \((2, -2)\), so:

$$ f(2) = -2 $$

Thus:

$$ (f \circ g)(-1) = -2 $$

Evaluate (f ∘ g)(4)

Using the Composite Functions knowledge point

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

From the graph of \(y = g(x)\), the point at \(x = 4\) is \((4, 1)\), so:

$$ g(4) = 1 $$

Now evaluate \(f(1)\). From the graph of \(y = f(x)\), the point at \(x = 1\) is \((1, -1)\), so:

$$ f(1) = -1 $$

Thus:

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

Answer:

Question a

\( (g \circ f)(-1) = 2 \)

Question b

\( (g \circ f)(0) = 4 \)

Question c

\( (f \circ g)(-1) = -2 \)

Question d

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