COMPOSITION OF FUNCTIONS FROM A GRAPH

Problem 1 :

Use the graph of f(x) and g(x) to find the composition of functions.

Solution:

a. f(g(2))

From the graphs given above, it is clear that the line at x = 2  is crossing the curve g(x) at the point 0.

The value of g(2) is 0.

f(g(2)) = f(0)

When x = 0, the vertical line crosses the curve f(x) at -3.

f(g(2)) = -3

b. [f ∘ g] (0)

From the graphs given above, it is clear that the line at x = 0  is crossing the curve g(x) at the point -2.

The value of g(0) is -2.

f(g(0)) = f(-2)

When x = -2, the vertical line crosses the curve f(x) at 5.

f(g(0)) = 5

c. g(f(2))

From the graphs given above, it is clear that the line at x = 2  is crossing the curve f(x) at the point -3.

The value of f(2) is -3.

g(f(2)) = g(-3)

When x = -3, the vertical line crosses the curve g(x) at -5.

g(f(2)) = -5

d. [g ∘ f] (-1)

From the graphs given above, it is clear that the line is crossing the curve f(x) at 0.

The value of f(-1) is 0.

g(f(-1)) = g(0)

When x = 0, the vertical line crosses the curve g(x) at -2.

g(f(-1)) = -2

e. g(g(4))

From the graphs given above, it is clear that the line is crossing the curve g(x) at 2.

The value of g(4) is 2.

g(g(4)) = g(2)

When x = 2, the vertical line crosses the curve g(x) at 0.

g(g(4)) = 0

f. [f ∘ f] (3)

From the given graph, it is clear that the line is crossing the curve f(x) the line at 0.

The value of f(3) is 0.

f(f(3)) = f(0)

When x = 0, the vertical line crosses the curve f(x) at -3.

f(f(3)) = -3

g. f(g(f(1)))

From the given graph, it is clear that the line is crossing the curve f(x) at -4.

The value of f(1) is -4.

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

When x = -4, the vertical line crosses the curve g(x) at -6.

f(g(-4)) = -6

When x = -6, the vertical line crosses the curve f(x) at -6.

f(g(f(1))) = 45

h. g(f(g(0)))

From the given graph, it is clear that the line is crossing the curve g(x) the vertical line at -2.

The value of g(0) is -2.

g(f(g(0))) = g(f(-2))

When x = -2, the vertical line crosses the curve f(x) at 5.

g(f(-2)) = 5

When x = 5, the vertical line crosses the curve g(x) at 3.

g(f(g(0))) = 3

Problem 2 :

Use the graph to determine the value of each composite function.

Solution:

a.  (h ∘ f) (3)

(h ∘ f) (3) = h(f(3))

From the given graph, it is clear that the line f(x) is crossing the vertical line at -4.

The value of f(3) is -4.

h(f(3)) = h(-4)

When x = -4, the vertical line crosses the curve h(x) at 5.

h(f(3)) = 5

b. (f ∘ g) (4)

(f ∘ g) (4) = f(g(4))

From the given graph, it is clear that the line is crossing the  curve g(x) at 2.

The value of g(4) is 2.

f(g(4)) = f(2)

When x = 2, the vertical line crosses the curve f(x) at -3.

f(g(4)) = -3

c. (f ∘ f) (-4)

(f ∘ f) (-4) = f(f(-4))

From the given graph, it is clear that the line is crossing f(x) the curve at -5.

The value of f(-4) is -5.

f(f(-4)) = f(-5)

When x = -5, the vertical line crosses the curve f(x) at -6.

f(f(-4)) = -6

d. (g ∘ g) (1)

(g ∘ g) (1) = g(g(1))

From the given graph, it is clear that the line is crossing the curve g(x) at 0.

The value of g(1) is 0.

g(g(1)) = g(0)

When x = 0, the vertical line crosses the curve g(x) at 2.

g(g(1)) = 2

e. (g ∘ h) (0)

(g ∘ h) (0) = g(h(0))

From the given graph, it is clear that the line h(x) is crossing the vertical line at 5.

The value of h(0) is 5.

g(h(0)) = g(5)

When x = 5, the vertical line crosses the curve g(x) at 0.

g(h(0)) = 0

Problem 3 :

Use the graphs to evaluate the expressions below.

Solution:

a. f(g(3))

From the given graph, it is clear that the line is crossing g(x) at 4.

The value of g(3) is 4.

f(g(3)) = f(4)

When x = 4, the vertical line crosses the curve f(x) at 0.

f(g(3)) = 0

b. f(g(1))

From the given graph, it is clear that the line is crossing the g(x) at 3.

The value of g(1) is 3.

f(g(1)) = f(3)

When x = 3, the vertical line crosses the curve f(x) at 2.

f(g(1)) = 2

c. g(f(1))

From the given graph, it is clear that the line is crossing f(x) at 3.

The value of f(1) is 3.

g(f(1)) = g(3)

When x = 3, the vertical line crosses the curve g(x) at 4.

g(f(1)) = 4

d. g(f(0))

From the given graph, it is clear that the line is crossing f(x) at 4.

The value of f(0) is 4.

g(f(0)) = g(4)

When x = 4, the vertical line crosses the curve g(x) at 5.

g(f(0)) = 5

e. f(f(5))

From the given graph, it is clear that the line is crossing at f(x) at 1.

The value of f(5) is 1.

f(f(5)) = f(1)

When x = 1, the vertical line crosses the curve f(x) at 3.

f(f(5)) = 3

f. f(f(4))

From the given graph, it is clear that the line is crossing f(x) at 0.

The value of f(4) is 0.

f(f(4)) = f(0)

When x = 0, the vertical line crosses the curve f(x) at 4.

f(f(4)) = 4

g. g(g(2))

From the given graph, it is clear that the line is crossing g(x) at 0.

The value of g(2) is 0.

g(g(2)) = g(0)

When x = 0, the vertical line crosses the curve g(x) at 2.

g(g(2)) = 2

h. g(g(0))

From the given graph, it is clear that the line is crossing g(x) the vertical line at 2.

The value of g(0) is 2.

g(g(0)) = g(2)

When x = 2, the vertical line crosses the curve g(x) at 0.

g(g(0)) = 0