Questions: (a) Find the inverse function of f(x) = 5x - 3. f^(-1)(x) = (x + 3) / 5 (b) The graphs of f and f^(-1) are symmetric with respect to the line defined by y =

(a) Find the inverse function of f(x) = 5x - 3. f^(-1)(x) = (x + 3) / 5 (b) The graphs of f and f^(-1) are symmetric with respect to the line defined by y =
Transcript text: (a) Find the inverse function of $f(x)=5 x-3$. \[ f^{-1}(x)=\frac{x+3}{5} \] (b) The graphs of $f$ and $f^{-1}$ are symmetric with respect to the line defined by $y=$ $\square$ Question Help: Video
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Solution

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Solution Steps

Solution Approach

(a) To find the inverse function of \( f(x) = 5x - 3 \), we need to solve for \( x \) in terms of \( y \) where \( y = f(x) \). Then, we express \( x \) as a function of \( y \).

(b) The graphs of \( f \) and \( f^{-1} \) are symmetric with respect to the line \( y = x \).

Step 1: Finding the Inverse Function

To find the inverse function of \( f(x) = 5x - 3 \), we set \( y = f(x) \), which gives us the equation \( y = 5x - 3 \). Rearranging this equation to solve for \( x \) yields: \[ x = \frac{y + 3}{5} \] Thus, the inverse function is: \[ f^{-1}(x) = \frac{x + 3}{5} \]

Step 2: Symmetry of the Graphs

The graphs of \( f \) and \( f^{-1} \) are symmetric with respect to the line defined by \( y = x \). This means that for any point \( (a, b) \) on the graph of \( f \), the point \( (b, a) \) will be on the graph of \( f^{-1} \).

Final Answer

The inverse function is \( f^{-1}(x) = \frac{x + 3}{5} \) and the line of symmetry is \( y = x \). Therefore, the answers are: \[ \boxed{f^{-1}(x) = \frac{x + 3}{5}} \] \[ \boxed{y = x} \]

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