How to shift input for testing shift invariance in a system?
Time invariance: changing $x$ by $x-x_0$ do not affect anything. 

Example 1: "Multiplying the argument by 2" operator
$$H(f(x))=f(2x)$$

Note the operator is very tricky: multiplying the argument by 2. 

For example:
$$H(sin(x))=sin(2x), H(x^2+1)=4x^2+1, H(e^x)=e^{2x}$$

The invariance is tested by comparing a shift in the signal vs a shift in the result of the operator:
$$H(f)|_{x-x_0}=H(f|_{x-x_0})$$

For the LHS:
$$H(f)|_{x-x_0}=f(2x)(x-x_0)=f(2(x-x_0))$$
The function is $f(2x)$, which is shifted by $x_0$.

For the RHS
$$H(f|_{x-x_0})=f(2x-x_0)$$
The function is $f(x-x_0)$, which is converted to $f(2x-x_0)$.