To find the inverse function [tex]\( f^{-1}(x) \)[/tex] given [tex]\( f(x) = \frac{5x - 1}{2} \)[/tex], we can follow these steps:
1. Write the function in the form of an equation:
[tex]\[ y = \frac{5x - 1}{2} \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse relation:
[tex]\[ x = \frac{5y - 1}{2} \][/tex]
3. Solve this new equation for [tex]\( y \)[/tex]:
[tex]\[ 2x = 5y - 1 \][/tex]
[tex]\[ 2x + 1 = 5y \][/tex]
[tex]\[ y = \frac{2x + 1}{5} \][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{2x + 1}{5} \][/tex]
Now, let's verify the result using a sample value to confirm the correctness.
### Verification
Choose a value for [tex]\( x \)[/tex], say [tex]\( x = 1 \)[/tex].
First, we calculate [tex]\( f(1) \)[/tex]:
[tex]\[ f(1) = \frac{5(1) - 1}{2} = \frac{5 - 1}{2} = \frac{4}{2} = 2 \][/tex]
Next, we substitute [tex]\( y = 2 \)[/tex] into our inverse function:
[tex]\[ f^{-1}(2) = \frac{2(2) + 1}{5} = \frac{4 + 1}{5} = \frac{5}{5} = 1 \][/tex]
Since [tex]\( f^{-1}(f(1)) = 1 \)[/tex], we verify that our inverse function is correct.
Therefore, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = \frac{2x + 1}{5} \][/tex]
