Let f (x) = x^2 + 1 and g (x) = x - 5. calculate (fog) (1) and (gof) (1)

Answer 1

Hi! I can surely help you with your composition of functions problem :)

First, what fog(1) means is: g is a function of f, evaluated at 1.

Another notation for fog(1) is: f(g(1))

What it means is that wherever there is an x in the f(x) function, replace that x with the entire function g(x). Simplify, then evaluate the result using x=1.

How do you do it? Thankfully, its simpler than its explanation! I always tell my students to do these problems from right to left: Put 1 into g, then put that answer into f.

The problem, part 1: Find fog(1)

The given info: f(x) = x² + 1 and g(x) = (x-5)

The steps:

1. Put the 1 into g(x) and simplify - or in other words, find g(1). Do this by replacing all xs in the g(x) equation with 1:
2. g(1) = 1-5 ⇒ g(1) = -4
3. Put g(1) into f(x) and simplify. Since we found that g(1)=-4 in step 1, we will compute f(-4) to find the final answer for fog(1):
4. f(g(1)) is what we are asked to find.
5. g(1) = -4, so with substitution, we will find f(-4)
6. f(x) = x² + 1, so to find f(-4), replace the x with -4
7. f(-4) = (-4)² + 1 ⇒ f(-4) = 16 + 1 ⇒ f(-4) = 17

Part 1 Answer: fog(1) = 17

The problem, part 2: find gof(1)

The functions: f(x) = x² + 1 and g(x) = (x - 5)

Execute from right to left: put the 1 into f, then put that answer into g.

evaluate f(1):

f(1) = 1² + 1 ⇒ f(1) = 2

Now put that answer into g:

gof(1) = g(f(1)) = g(2)

g(2) = 2 - 5 ⇒ g(2) = -3

Since gof(1) = g(f(1)) = g(2), gof(1) = -3

Part 2 Answer: gof(1) = -3

Answer 2

f(x) = x² + 1

g(x) = x - 5

f ° g(x) = (x - 5)² + 1

f ° g(1) = (1 - 5)² + 1

= (-4)² + 1

= 17

g º f(x) = (x² + 1) - 5

g º f(1) = (1² + 1) - 5

= 2 - 5

= -3