Solve for the equations

Question

I have no idea what to do here.

If g(x) = 5x²+7 and f(x)= x²+1, then what is g(f(x)) and f(g(x))?

What is the value of both equations if x =2?

Answer 1

When given two functions, we will substitute one for x into the other.

So, if g(x)= 5x²+7 and f(x)= x²+1

Then, to solve for g(f(x)) we will take the f function and use it in place of x in the g function.

5x²+7

5(f)²+7

g(f(x))= 5(x²+1)²+7

To solve for f(g(x)) we substitute g in for x into the f function

x²+1

(g)²+1

f(g(x))= (5x²+7)²+1

Then it asks for the solutions to these new functions when x = 2

We do this by now substituting 2 in for x in these functions

Follow the order of operations or PEMDAS

g(f((2))= 5(x²+1)²+7

5((2)²+1)²+7

5(4+1)²+7

5(5)²+7

5(25)+7

125+7

132

f(g(2))= (5x²+7)²+1

(5(2)²+7)²+1

(5(4)+7)²+1

(20+7)²+1

27²+1

729+1

730

Answer 2

g(x) = 5x²+7 and f(x)= x²+1.

Before solving for g(f(x)) and f(g(x)), lets understand what these are. So, these are called composite functions in which you can input one function into another function. So, we know we have two functions named f(x) and g(x). When we say g(f(x)), you notice that f(x) replace the x part inside of the g(x). Since the f(x) replace the x, it means we also replace the x of the g(x) function with whatever f(x) is. We know f(x) is x²+1 which replaces the x inside of g(x) so it becomes:

g(f(x)) = 5(x²+1)² + 7 Notice we just replaced the x of g(x) with what f(x) is. And now we can just simplify it.

= 5(x²+1)(x²+1) + 7

= 5(x⁴ + x² + x² + 1) + 7

= 5(x⁴ + 2x² + 1) + 7

= 5x⁴ + 10x² + 5 + 7

g(f(x)) = 5x⁴ + 10x² + 12

In the same way, we can solve for f(g(x)) but now we replace the x of f(x) with g(x) this time.

f(g(x)) = (5x² + 7)² + 1 Now, simplify it

= (5x² + 7)(5x² + 7) + 1

= (25x⁴ + 35x² + 35x² + 49) + 1

= 25x⁴ + 70x² + 49 + 1

f(g(x)) = 25x⁴ + 70x² + 50

Now, to solve for the value of both equations if x = 2, we can just plug in 2 for x into both of our equations.

The first equation that we solved for:

g(f(x)) = 5x⁴ + 10x² + 12 We plug in 2 into x

g(f(2)) = 5(2)⁴ + 10(2)² + 12

= 5(16) + 10(4) + 12

= 80 + 40 + 12

g(f(2)) = 132

The second equation that we solved for:

f(g(x)) = 25x⁴ + 70x² + 50 We plug in 2 into x

f(g(x)) = 25(2)⁴ + 70(2)² + 50

= 25(16) + 70(4) + 50

= 400 + 280 + 50

f(g(2)) = 730

Answer 3

We have g(x) = 5x² +7, f(x) = x² +1

f(g(x)) means we have to replace every x , in f(x) with g(x)

so f(g(x))= (g(x))²+1=(5x²+7)²+1=25x⁴+70x²+49+1=25x⁴+70x²+50

Also g(f(x))=5(f(x))²+7=5(x²+1)²+7=5(x⁴+2x²+1)+7=5x⁴+10x²+5+7=5x⁴+10x²+12

so f(g(2))=25×(2)⁴+70×(2)²+50=730

& g(f(2))=5×(2)⁴+10×(2)²+12=132

Answer 4

Hello, thank you for taking the time to post your question!

These are examples of composite functions so if you’re trying to find g(f(x)) that means that you want to substitute in f(x) back into g(x) for the value of x. For these functions that would mean

g(f(x)) = 5*(f(x))^2 + 7

= 5(x^2 + 1)^2 + 7

Similarly f(g(x)) would be

f(g(x)) = (g(x))^2 + 1

=(5x^2 + 7)^2 + 1

To find the value of both equations then you would just plug in x = 2 for the value of x. When I do that using a calculator I find the values to be

g(f(2)) = 132 and

f(g(2)) = 730

Hopefully that gets you moving in the right direction! Feel free to reach out if you have any questions beyond that :)

Answer 5

Answer 6

g(x) = 5x^2 +7, f(x) = x^2 +1

g(f(x)) = 5(f(x)^2 +7 = 5(x^2+1)^2 +7 = 5(x^4 +2x^2+1) +7 = 5x^4 +10x^2 +12

f(g(x)) = (g(x))^2 +1 = (5x^2+7)^2 +1 = 25x^4 +70x^2 +50

g(f(2)) = 5(2^4) + 10(2^2) +12 = 80+40+12 = 132

f(g(2) = 25(2^4) + 70(2^2) +50 = 400 +280+50 = 730