Construct a function that passes through the origin with a constant slope of 1 , with removable discontinuities at $x=-5$ and $x=3$.
Enclose numerators and denominators in parentheses. For example, $(a-b) /(1+n)$.
\[
f(x)=
\]
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Step 1 ：Construct a function that passes through the origin with a constant slope of 1, with removable discontinuities at $x=-5$ and $x=3$.

Step 2 ：The function that passes through the origin with a constant slope of 1 is simply $f(x) = x$.

Step 3 ：We need to introduce removable discontinuities at $x=-5$ and $x=3$. We can do this by multiplying $f(x)$ by a function that is undefined at these points.

Step 4 ：A simple function that is undefined at a point $a$ is $\frac{1}{x-a}$. Therefore, we can introduce the discontinuities by multiplying $f(x)$ by $\frac{1}{(x+5)(x-3)}$.

Step 5 ：This will also change the values of $f(x)$ at all other points. To avoid this, we can add 1 to the denominator, which makes the function equal to 1 at all points except $x=-5$ and $x=3$, where it is undefined.

Step 6 ：Therefore, the function we are looking for is $f(x) = x \cdot \frac{1}{1 + \frac{1}{(x+5)(x-3)}}$.

Step 7 ：Final Answer: The function that passes through the origin with a constant slope of 1 and has removable discontinuities at $x=-5$ and $x=3$ is $\boxed{f(x) = \frac{x}{1 + \frac{1}{(x+5)(x-3)}}}$.