Given the functions f(x) = 2x + 1 and g(x) = x - 3, determine an equation for (f ∘ g)(x) and (g ∘ f)(x).
Determine f(g(5)) and g(f(5)).
Determine all values of x for which f(g(x)) = g(f(x)
(f ∘ g)(x) = f(g) = 2g+1 = 2(x-3)+1 = 2x-5
f(g(5)) = 2*5-5 = 5
or,
f(g(5)) = f(5-3) = f(2) = 2*2+1 = 5
do the same steps for (g∘f)(x)
then, when you have (g∘f) as a function of x, just equate 2x+1 = (g∘f)(x)
Since they are both linear, there will be at most one solution, unless they are the same line.
(f ∘ g)(x) can be computed by substituting g(x) into f(x):
(f ∘ g)(x) = f(g(x)) = 2(g(x)) + 1 = 2(x - 3) + 1 = 2x - 5.
(g ∘ f)(x) can be computed by substituting f(x) into g(x):
(g ∘ f)(x) = g(f(x)) = f(x) - 3 = 2x + 1 - 3 = 2x - 2.
To find f(g(5)), we need to substitute 5 into g(x) and then substitute the result into f(x):
g(5) = 5 - 3 = 2.
f(g(5)) = f(2) = 2(2) + 1 = 4 + 1 = 5.
To find g(f(5)), we need to substitute 5 into f(x) and then substitute the result into g(x):
f(5) = 2(5) + 1 = 10 + 1 = 11.
g(f(5)) = g(11) = 11 - 3 = 8.
To find all values of x for which f(g(x)) = g(f(x)), we can set the two expressions equal to each other and solve for x:
2x - 5 = 2x - 2.
Subtracting 2x from both sides, we get:
-5 = -2.
This equation has no solution, so there are no values of x for which f(g(x)) = g(f(x)).
To find the composition of functions (f ∘ g)(x) and (g ∘ f)(x), we substitute the expression for g(x) into f(x) and vice versa:
1. (f ∘ g)(x):
Substitute g(x) into f(x):
(f ∘ g)(x) = f(g(x)) = 2(g(x)) + 1 = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5
2. (g ∘ f)(x):
Substitute f(x) into g(x):
(g ∘ f)(x) = g(f(x)) = (f(x)) - 3 = (2x + 1) - 3 = 2x + 1 - 3 = 2x - 2
Next, we'll find the values of f(g(5)) and g(f(5)):
3. f(g(5)):
Substitute g(5) into f(x):
f(g(5)) = f(5 - 3) = f(2) = 2(2) + 1 = 4 + 1 = 5
4. g(f(5)):
Substitute f(5) into g(x):
g(f(5)) = g(2(5) + 1) = g(10 + 1) = g(11) = 11 - 3 = 8
Finally, we'll determine all values of x for which f(g(x)) = g(f(x)):
5. f(g(x)):
Substitute g(x) into f(x):
f(g(x)) = 2(g(x)) + 1 = 2(x - 3) + 1 = 2x - 6 + 1 = 2x - 5
6. g(f(x)):
Substitute f(x) into g(x):
g(f(x)) = (f(x)) - 3 = (2x + 1) - 3 = 2x + 1 - 3 = 2x - 2
To find the values of x for which f(g(x)) = g(f(x)), set the two expressions equal to each other and solve for x:
2x - 5 = 2x - 2
-5 = -2
The equation has no valid solutions.
To find the equation for (f ∘ g)(x), you need to substitute g(x) into f(x). This means that you replace x in the function f(x) with g(x).
Starting with f(x) = 2x + 1, substitute g(x) = x - 3:
(f ∘ g)(x) = f(g(x))
= f(x - 3)
= 2(x - 3) + 1
= 2x - 6 + 1
= 2x - 5
Therefore, the equation for (f ∘ g)(x) is 2x - 5.
To find the equation for (g ∘ f)(x), you need to substitute f(x) into g(x). This means you replace x in the function g(x) with f(x).
Starting with g(x) = x - 3, substitute f(x) = 2x + 1:
(g ∘ f)(x) = g(f(x))
= g(2x + 1)
= (2x + 1) - 3
= 2x + 1 - 3
= 2x - 2
The equation for (g ∘ f)(x) is 2x - 2.
To find f(g(5)), you first plug 5 into g(x):
g(5) = 5 - 3
= 2
Then you substitute the result into f(x):
f(g(5)) = f(2)
= 2(2) + 1
= 4 + 1
= 5
Therefore, f(g(5)) = 5.
To find g(f(5)), you first substitute 5 into f(x):
f(5) = 2(5) + 1
= 10 + 1
= 11
Then you substitute the result into g(x):
g(f(5)) = g(11)
= 11 - 3
= 8
Therefore, g(f(5)) = 8.
To determine all values of x for which f(g(x)) = g(f(x)), you need to equate the two expressions and solve for x:
f(g(x)) = g(f(x))
Substituting the actual functions:
2(g(x)) + 1 = g(2x + 1)
Substituting g(x) = x - 3:
2(x - 3) + 1 = (2x + 1) - 3
Expanding both sides:
2x - 6 + 1 = 2x - 2
Simplifying:
2x - 5 = 2x - 2
Subtracting 2x from both sides:
-5 = -2
This equation has no valid solution. Therefore, there are no values of x for which f(g(x)) = g(f(x)).