To determine whether the function [tex]\( f(x) = 8x + 7 \)[/tex] has an inverse, we need to check whether it is a one-to-one function. A one-to-one function means each output is produced by exactly one input.
For a linear function of the form [tex]\( f(x) = ax + b \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants, the function is one-to-one if [tex]\( a \neq 0 \)[/tex].
Here,
[tex]\[ f(x) = 8x + 7 \][/tex]
The coefficient [tex]\( a \)[/tex] is 8, which is not equal to zero. Therefore, the function [tex]\( f(x) = 8x + 7 \)[/tex] is indeed one-to-one and thus has an inverse.
To find the inverse function, we follow these steps:
1. Rewrite the function [tex]\( f(x) = y \)[/tex].
[tex]\[ y = 8x + 7 \][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to begin finding the inverse.
[tex]\[ x = 8y + 7 \][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
[tex]\[ x - 7 = 8y \][/tex]
[tex]\[ y = \frac{x - 7}{8} \][/tex]
4. Replace [tex]\( y \)[/tex] with [tex]\( f^{-1}(x) \)[/tex] to denote the inverse function.
[tex]\[ f^{-1}(x) = \frac{x - 7}{8} \][/tex]
So, the inverse function is:
[tex]\[ f^{-1}(x) = \frac{x - 7}{8} \][/tex]
Therefore:
- Yes, [tex]\( f \)[/tex] does have an inverse.
- The inverse function is [tex]\( f^{-1}(x) = \frac{x - 7}{8} \)[/tex].
