To determine whether \( k(x) \) and \( j(x) \) are inverse functions, we need to check if \( k(j(x)) = x \) and \( j(k(x)) = x \).
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Finding \( k(j(x)) \):
- Substitute \( j(x) = 4x + 6 \) into \( k(x) \): \[ k(j(x)) = k(4x + 6) = 7 - \frac{(4x + 6)}{4} - 2(4x + 6) \]
Simplifying this:
- First calculate \( \frac{(4x + 6)}{4} = x + \frac{3}{2} \).
- Now plug this back into \( k \): \[ k(j(x)) = 7 - (x + \frac{3}{2}) - 2(4x + 6) \] \[ = 7 - x - \frac{3}{2} - (8x + 12) \] \[ = 7 - x - 8x - 12 - \frac{3}{2} = 7 - 12 - \frac{3}{2} - 9x \] The simplification show that it does not equal \( x \).
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Finding \( j(k(x)) \):
- Substitute \( k(x) = 7 - \frac{x}{4 - 2x} \) into \( j(x) \): \[ j(k(x)) = j(7 - \frac{x}{4 - 2x}) = 4(7 - \frac{x}{4 - 2x}) + 6 \]
- Simplifying, we will arrive at a complex polynomial, not guaranteed to be equal to \( x \).
Since we can derive from above that \( k(j(x)) \neq x \) for at least one substitution, we can conclude that:
The functions \( k(x) \) and \( j(x) \) are not inverse functions.
Thus, the correct answer is: False.