Describe step by step transformations of the function: \[ f(x)=-2|\ln (x)+1|+1 \]
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First off, let’s break down our function step by step! We start with the natural logarithm function \( \ln(x) \), which is defined for \( x > 0 \). As we take \( \ln(x) \), it goes from negative infinity to positive infinity as \( x \) ranges from 0 to \( \infty \). Now, move on to the transformation! First, adding 1 inside the absolute value, we have \( |\ln(x) + 1| \). This shifts the graph left by \( e^{-1} \) on our natural log curve. Next, multiplying by -2 flips the graph vertically and stretches it vertically, making it concave down. After that, we add 1 at the end, which shifts the entire graph up by 1 unit. So our function not only reflects, but also “bobs” up, creating a fabulous new dance on the Cartesian plane! All these transformations go hand in hand to create an interesting graph. It starts below the horizontal line \( y = 1 \) for values of \( \ln(x) + 1 < 0 \) and peaks as \( |\ln(x) + 1| \) becomes large, creating intriguing local maxima depending on your input \( x \). Don't forget that it only exists for positive values of \( x \); beyond that, it's like trying to enter a party without an invitation!
