The graph of y = f (x) is shifted left by a, reflected in the y-axis and finally shifted right by a. What is the equation of the new graph?

Question

I know that a shift left by a means f(x+a). Then a y-axis reflection gives f(-(x+a)). Finally, shifting right again gives f(-x-a-a) = f(-2a-x). However, the answers stipulate f(2a-x), as shown. What am I doing wrong?

Answer 1

I don't understand how the book is defining its transformations so I'm ignoring it. As far as your steps go, you're just making a mistake in the second step.

Step 1 - shift left by $a$: to do this, we replace $x$ with $x+a$. So

$$f(x) \longrightarrow f(x+a)$$

Step 2 - reflect in $y$-axis: to do this, we replace $x$ with $-x$. So

$$f(x+a) \longrightarrow f(-x+a)$$

Step 3 - shift right by $a$: to do this, we replace $x$ with $x-a$. So

$$f(-x+a) \longrightarrow f\left[-(x-a)+a\right]=f(2a-x)$$

Answer 2

When reflecting across the $y$-axis, only the variables $x$ must be negated, not the whole argument to $f$. So it should be $f(-x+a)$.