If the graph of \( f(x)=\frac{1}{x^{2}} \) is shifted up 2 units and to the left 4 units, then the result is: \( g(x)=\frac{1}{(x+4)^{2}}+2 \) \( g(x)=\frac{1}{x^{2}+4}+2 \) \( g(x)=\frac{1}{(x-4)^{2}}+2 \) \( g(x)=\frac{1}{x^{2}-4}-2 \)

Shifting the graph of \( f(x) = \frac{1}{x^2} \) up 2 units means you add 2 to the entire function. Shifting it to the left 4 units requires you to adjust the input \( x \) by replacing \( x \) with \( x + 4 \). Thus, you get: \[ g(x) = \frac{1}{(x + 4)^2} + 2 \] So the correct choice is \( g(x) = \frac{1}{(x+4)^{2}}+2 \). Understanding this transformation is like playing with a digital image – you can move it around in space by adding or adjusting dimensions, which keeps the essence intact, yet creates a new perspective! When shifting functions, a common mistake is to mix the directions. Remember that moving left involves adding to \( x \) (because you’re "pulling" the input towards the left), while moving up simply adds to the output. So keep an eye on your signs and operations, as they really do change the resulting graph!