Answer:
Last is correct
Step-by-step explanation:
To determine which of the given statements is true for \( f(x) = 5x + 1 \) and its inverse, we need to analyze the function and its inverse.
1. **Original Function**: \( f(x) = 5x + 1 \)
- This is a linear function with a slope of 5 and a y-intercept of 1.
- A linear function is a one-to-one function, meaning that each input \( x \) maps to exactly one output \( f(x) \). Therefore, \( f(x) = 5x + 1 \) is indeed a function.
2. **Inverse Function**:
- To find the inverse, we swap \( x \) and \( y \) in the equation \( y = 5x + 1 \) and solve for \( y \):
\[ x = 5y + 1 \]
\[ x - 1 = 5y \]
\[ y = \frac{x - 1}{5} \]
- The inverse function is \( f^{-1}(x) = \frac{x - 1}{5} \).
- The inverse function \( f^{-1}(x) = \frac{x - 1}{5} \) is also a linear function, and linear functions are always one-to-one, meaning that each input \( x \) maps to exactly one output \( f^{-1}(x) \). Therefore, \( f^{-1}(x) = \frac{x - 1}{5} \) is a function.
Given this analysis, the correct statement is:
- Both are functions
So, the true relation for \( f(x) = 5x + 1 \) is that both the equation and its inverse are functions.
