HOW TO EVALUATE COMPOSITION OF FUNCTION FROM GRAPH

Problem 1 :

The first of the two graphs shows two functions 𝑓 𝑎𝑛𝑑 𝑔. The second shows two functions ℎ 𝑎𝑛𝑑 𝑘. Use the graphs to compute the following:

Solution :

1) (𝑔 ◦ 𝑓)(−4) = g[f(-4)]

To get the value of f(-4), we have to draw the vertical line through x = -4.

The vertical line is intersecting the graph f at 0. So, the value of f(-4) is 0.

(𝑔 ◦ 𝑓)(−4) = g[0]

Now, we have to observe the point of intersection of the vertical line drawn at x = 0 to the function g. At -3, the vertical line is intersecting the graph g. So,

(𝑔 ◦ 𝑓)(−4) = -3

2)  (𝑓 ◦ 𝑔)(3) = f[g(3)] = f[0] ==> 2

3) (𝑓 ◦ 𝑓)(−2) = f[f(-2)] = f[3] ==> 1

4) (𝑔 ◦ 𝑔)(3) = g[g(3)] = g[0] ==> -3

5) (𝑔 ◦ 𝑓)(−5) = g[f(-5)] = g[-3] ==> -5

6) (𝑔 ◦ 𝑓)(−3) = g[f(-3)] = g[1.5] ==> -1.5

7) (ℎ ◦ 𝑘)(0) = h[k(0)] = h[4] ==>  4

8) (ℎ ◦ 𝑘)(−1) = h[k(-1)] = h[3] ==> 3.5

9) (ℎ ◦ 𝑘)(2) = h[k(2)] = h[0] ==> 2

10) (ℎ ◦ 𝑘)(−3) = h[k(-3)] = h[-5] ==> -0.5

11) (𝑘 ◦ ℎ)(0) = k[h(0)] = k[2] ==> 0

12) (𝑘 ◦ ℎ)(2) = k[h(2)] = k[3] ==> -5

13) (𝑘 ◦ ℎ)(−4) = k[h(-4)] = k[0] ==> 4

14) (𝑘 ◦ ℎ)(−2) = k[h(-2)] = k[1] ==> 3

Problem 2 :

Refer to the graph to complete the statements below.

a) (f + g)(-3) = ______

b) (f · g)(2) = ______

c) (f/g)(-1) = ______

d) (f ∘ g)(3) = ______

e) g^-1(-4) = ______

f) Evaluate (f ∘ f)(2) ______

g) Evaluate g(f(g(1))) ______

h) State the domain of f + g _____

i) State the domain of f/g. ______

j) Evaluate (f(3))³ - 4g(-2) ______

k) For what value(s) is f(x) = 3? _______

Solution:

a) (f + g)(-3)

(f + g)(-3) = f(-3) + g(-3)

To evaluate f(-3) and g(-3), we have to draw the vertical lines at x = -3. The line is intersecting the curve f(x) and g(x) at 4 and -5 respectively.

(f + g)(-3) = f(-3) + g(-3)

= 4 + (- 5)

(f + g)(-3)  = -1

b) (f · g)(2)

(f · g)(2) = f(2) · g(2)

To evaluate f(2) and g(2), we have to draw the vertical lines at x = 2. The line is intersecting the curve f(x) and g(x) at 3 and 0 respectively.

(f · g)(2) = f(2) · g(2)

= 3 · 0

(f · g)(2) = 0

c) (f/g)(-1)

(f/g)(-1) = f(-1) / g(-1)

To evaluate f(-1) and g(-1), we have to draw the vertical lines at x = -1. The line is intersecting the curve f(x) and g(x) at 2 and -3 respectively.

(f/g)(-1) = f(-1) / g(-1)

(f/g)(-1) = 2/(-3)

d) (f ∘ g)(3) 

(f ∘ g)(3) = f [g(3)]

= f [1]

(f ∘ g)(3) = 4

e)  g^-1(-4) 

f(x) = y, then f^-1(y) = x

Let g^-1(-4) = x

-4 = g(x)

for what value of x, the value of y will be -4. So x = -3.

f) (f ∘ f)(2) 

(f ∘ f)(2) = f [f(2)]

= f [3]

(f ∘ f)(2) = 2

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

g(f(g(1))) = f [g(1)]

= g [f (-1)]

= g(2)

= 0

h) State the domain of f + g _____

Domain of f(x) = [-6, 7]

Domain of g(x) = [-3, 4]

[-6, 7] n [-3, 4] = [-3, 4]

[-6, 7] n [-3, 4] = [-3, 4]

So, the domain of (f + g)(x) is [-3, 4].

i) State the domain of f/g. ______

From the graph above, domain of (f/g)(x) = [-3, 1] u (3, 4]

j) (f(3))³ - 4g(-2) 

(f(3))³ - 4g(-2) = (2)³ - 4(-4)

= 8 + 16

= 24

k) For what value(s) is f(x) = 3? _______

When drawing the horizontal line at y = 3, it intersects the curve f(x) at -2, 0 and 2.

So, the respective x values are -2, 0 and 2.