To determine whether the function [tex]\( f(x) = \frac{4x + 5}{5x + 7} \)[/tex] has an inverse, we can follow these steps:
1. Check if the function can be inverted: A function [tex]\( f(x) \)[/tex] has an inverse if it is one-to-one (bijective). For rational functions like this one, we typically find the inverse function by solving [tex]\( y = f(x) \)[/tex] for [tex]\( x \)[/tex].
2. Express y in terms of x: Start by expressing the given function in a general form:
[tex]\[
y = \frac{4x + 5}{5x + 7}
\][/tex]
3. Solve for x: To find the inverse, solve this equation for [tex]\( x \)[/tex]. Begin by clearing the fraction by multiplying both sides by [tex]\( 5x + 7 \)[/tex]:
[tex]\[
y(5x + 7) = 4x + 5
\][/tex]
4. Distribute y:
[tex]\[
5xy + 7y = 4x + 5
\][/tex]
5. Group the terms involving x on one side and the constant terms on the other side:
[tex]\[
5xy - 4x = 5 - 7y
\][/tex]
6. Factor out x:
[tex]\[
x(5y - 4) = 5 - 7y
\][/tex]
7. Solve for x by dividing both sides by [tex]\( 5y - 4 \)[/tex]:
[tex]\[
x = \frac{5 - 7y}{5y - 4}
\][/tex]
8. Replace y by x to get the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
f^{-1}(x) = \frac{5 - 7x}{5x - 4}
\][/tex]
Therefore, the function [tex]\( f(x) = \frac{4x + 5}{5x + 7} \)[/tex] does indeed have an inverse, and the inverse function is given by:
[tex]\[
f^{-1}(x) = \frac{5 - 7x}{5x - 4}
\][/tex]
