For the following function, a) determine whether it is one-to-one; ) if it is one-to-one, find its inverse fun

Is the given function a one-to-one function? Select the correct choice below and, if necessary, fill in the as
A. Yes, the function is one-to-one. The inverse function is
(Simplify your answer)

Ask by Colon Morrison. in the United States

Nov 24,2024

Question

For the following function, a) determine whether it is one-to-one; ) if it is one-to-one, find its inverse fun

Is the given function a one-to-one function? Select the correct choice below and, if necessary, fill in the as
A. Yes, the function is one-to-one. The inverse function is
(Simplify your answer)

Ask by Colon Morrison. in the United States

Nov 24,2024

UpStudy AI · Accepted Answer

To determine whether the function $$ f(x) = x - 6 $$ is one-to-one, we can use the definition of a one-to-one function: a function is one-to-one if it never assigns the same value to two different domain elements.

### a) Is the function one-to-one?

1. **Check if $$ f(a) = f(b) $$ implies $$ a = b $$**:
   $$
   f(a) = a - 6
   $$
   $$
   f(b) = b - 6
   $$
   If $$ f(a) = f(b) $$, then:
   $$
   a - 6 = b - 6
   $$
   Adding 6 to both sides gives:
   $$
   a = b
   $$
   Since $$ a = b $$ whenever $$ f(a) = f(b) $$, the function is one-to-one.

### b) Find the inverse function

To find the inverse function $$ f^{-1}(x) $$, we start by replacing $$ f(x) $$ with $$ y $$:
$$
y = x - 6
$$
Next, we solve for $$ x $$:
$$
y + 6 = x
$$
Now, we express $$ x $$ in terms of $$ y $$:
$$
x = y + 6
$$
To write the inverse function, we switch $$ x $$ and $$ y $$:
$$
f^{-1}(x) = x + 6
$$

### Conclusion

A. Yes, the function is one-to-one. The inverse function is $$ f^{-1}(x) = x + 6 $$.
A. Yes, the function is one-to-one. The inverse function is $$ f^{-1}(x) = x + 6 $$.

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