Determine algebraically if f(x) =(7x-2) / (4). And g(x) = (4x+2) / (7) are inverse functions. Show how you know,

I do not understand this type of problem i have a test on these tmrw and need some help with how to figure these out pls help.

Alternate way:

find inverse of f(x) to see if you get g(x)
let y = f(x) = (7x-2)/4
the inverse is :
x = (7y-2)/4
4x = 7y - 2
4x + 2 = 7y
y = (4x+2)/7 which is the given g(x)

so yes, they are inverses of each other.

f and g are inverse functions if

f(g(x)) = g(f(x)) = x

f(g) = (7g-2)/4 = (4x+2-2)/4 = x
g(f) = (4f+2)/7 = (7x-2+2)/7 = x

Well, let's see if we can figure this out together, even if I might not be the most helpful when it comes to algebra!

To determine if two functions are inverses of each other, we can follow a couple of steps:

Step 1: Set f(x) equal to y:
y = (7x - 2) / 4

Step 2: Swap x and y:
x = (7y - 2) / 4

Step 3: Solve for y:
4x = 7y - 2
4x + 2 = 7y
y = (4x + 2) / 7

Now we compare this result to g(x):

g(x) = (4x + 2) / 7

It seems like we ended up with the same expression for y in both cases. So, f(x) and g(x) might be inverse functions!

To double-check, we can perform the same process in reverse:

Step 1: Set g(x) equal to y:
y = (4x + 2) / 7

Step 2: Swap x and y:
x = (4y + 2) / 7

Step 3: Solve for y:
7x = 4y + 2
7x - 2 = 4y
y = (7x - 2) / 4

And we have ended up with the same expression for y as f(x).

So, based on our algebraic work, it appears that f(x) and g(x) are inverse functions!

Remember, it's always good to check your answers and work through the steps carefully. Good luck on your test!

Is this the answer or should i go further? Sorry im very lost

To determine if two functions, f(x) and g(x), are inverse functions, we need to follow two steps:

Step 1: Calculate the composition of f(g(x)) and g(f(x)).
Step 2: Simplify the results from Step 1 and compare them to the original input, x.

Let's start with Step 1:

1. Calculate f(g(x)):
Replace g(x) in f(x) with the expression (4x + 2) / 7:
f(g(x)) = f((4x + 2) / 7)
Substitute (4x + 2) / 7 into f(x) and simplify:
f(g(x)) = [7((4x + 2) / 7) - 2] / 4
= (28x + 14 - 2) / 28
= (28x + 12) / 28
= 7x/7 + 12/28
= x + 6/14
= x + 3/7

2. Calculate g(f(x)):
Replace f(x) in g(x) with the expression (7x - 2) / 4:
g(f(x)) = g((7x - 2) / 4)
Substitute (7x - 2) / 4 into g(x) and simplify:
g(f(x)) = [4((7x - 2) / 4) + 2] / 7
= (28x - 8 + 2) / 7
= (28x - 6) / 7
= 4x/7 - 6/7

Now, let's move to Step 2:

Compare the simplified expressions from Step 1 to the original input, x:

For f(g(x)), the simplified expression is x + 3/7.
For g(f(x)), the simplified expression is 4x/7 - 6/7.

If f(x) and g(x) are inverse functions, then f(g(x)) and g(f(x)) should simplify down to x.

Let's see if this is true:

For f(g(x)): x + 3/7 simplifies to x, which is the original input. Therefore, the first condition holds.

For g(f(x)): 4x/7 - 6/7 simplifies to 4x/7 - 6/7, which is not equal to x. Therefore, the second condition does not hold.

Since the two functions fail to satisfy both conditions, f(x) and g(x) are not inverse functions.

Remember to practice more example problems to become comfortable with the concept. Good luck with your test!