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 lower curve):

$$ f(-1) = 0 $$

Now evaluate \(g(0)\) from the graph of \(y = g(x)\) (the upper curve):

$$ g(0) = 3 $$

Therefore:

$$ (g \circ f)(-1) = 3 $$

Evaluate (g ∘ f)(0)

Using the Composite Functions knowledge point

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

From the graph of \(y = f(x)\):

$$ f(0) = -1 $$

Now evaluate \(g(-1)\) from the graph of \(y = g(x)\):

$$ g(-1) = 4 $$

Therefore:

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

Evaluate (f ∘ g)(-1)

Using the Composite Functions knowledge point

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

From the graph of \(y = g(x)\):

$$ g(-1) = 4 $$

Now evaluate \(f(4)\) from the graph of \(y = f(x)\):

$$ f(4) = -3 $$

Therefore:

$$ (f \circ g)(-1) = -3 $$

Evaluate (f ∘ g)(4)

Using the Composite Functions knowledge point

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

From the graph of \(y = g(x)\):

$$ g(4) = 1 $$

Now evaluate \(f(1)\) from the graph of \(y = f(x)\):

$$ f(1) = -2 $$

Therefore:

$$ (f \circ g)(4) = -2 $$

Answer:

(a) \((g \circ f)(-1) = 3\)

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

(c) \((f \circ g)(-1) = -3\)

(d) \((f \circ g)(4) = -2\)