Questions: Find f(g(x)) and g(f(x)) and determine whether the pair of functions f and g are inverses of each other. f(x)=6x-7 and g(x)=(x+6)/7

Transcript text: Find $f(g(x))$ and $g(f(x))$ and determine whether the pair of functions $f$ and $g$ are inverses of each other. \[ f(x)=6 x-7 \text { and } g(x)=\frac{x+6}{7} \]

Solution

Solution Steps

Step 1: Compute $f(g(x))$

To find $f(g(x))$, substitute $g(x) = \frac{x + 6}{7}$ into $f(x) = 6x - 7$. This gives $f(g(x)) = 6\left(\frac{x + 6}{7}\right) - 7$, which simplifies to $f(g(x)) = 4.14$.

Step 2: Compute $g(f(x))$

To find $g(f(x))$, substitute $f(x) = 6x - 7$ into $g(x) = \frac{x + 6}{7}$. This gives $g(f(x)) = \frac{(6x - 7) + 6}{7}$, which simplifies to $g(f(x)) = 5.86$.

Step 3: Determine if $f$ and $g$ are inverses of each other

Since $f(g(x)) \neq x$ or $g(f(x)) \neq x$ for some $x$, $f$ and $g$ are not inverses of each other.