Find the inverse function of y=
x - 7 divided by 4

f⁻¹(x) = 4x + 7

Step-by-step explanation:

To get the inverse of a given function, first isolate the x term of the equation by using the inverse operations on both sides to cancel them out:

y = (x - 7) / 4

×4 ×4

(multiplicative property of equality)

4y = x - 7

+7 +7

(addition property of equality)

4y + 7 = x

(symmetric property)

x = 4y + 7

Then swap the x, and y variables to get the inve se / your inverse function will be in terms of the equation you started with. Given that y = f(x), f(x) = (x - 7) / 4.

x = 4y + 7 → y = 4x + 7 → f⁻¹(x) = 4x + 7 .

Also keep in mind that for a function to be an inverse function of another, it must satisfy this composite rule:

f( f⁻¹(x) ) = x and f⁻¹( f(x) ) = x .

We can verify that they are inverses by substituting both of our equations into this rule:

f( f⁻¹(x) ) = 4 ( f⁻¹(x) ) + 7 = 4 ( ( x - 7 ) / 4 ) + 7 = ( x - 7 ) + 7 = x ✓

f⁻¹( f ( x ) ) = ( f ( x ) - 7 ) / 4 = ( 4x + 7 - 7 ) / 4 = 4x / 4 = x ✓

Since this matches the rule,

(x - 7) / 4 and 4x + 7 are true inverses.