1. The problem asks to sketch the graphs of two functions: a) $f(x) = x + |x|$ and b) $f(x) = |x + 2|$. 2. Important rules: - The absolute value function $|x|$ equals $x$ if $x \geq 0$ and $-x$ if $x < 0$. - This piecewise definition helps us rewrite and analyze the functions. 3. For a) $f(x) = x + |x|$: - If $x \geq 0$, then $|x| = x$, so $f(x) = x + x = 2x$. - If $x < 0$, then $|x| = -x$, so $f(x) = x + (-x) = 0$. 4. So the function is piecewise: $$ f(x) = \begin{cases} 2x & x \geq 0 \\ 0 & x < 0 \end{cases} $$ - This means the graph is the line $y=2x$ for $x \geq 0$ and the $x$-axis (line $y=0$) for $x < 0$. 5. For b) $f(x) = |x + 2|$: - The absolute value function shifts the graph of $|x|$ left by 2 units. - It is piecewise: $$ f(x) = \begin{cases} x + 2 & x + 2 \geq 0 \\ -(x + 2) & x + 2 < 0 \end{cases} = \begin{cases} x + 2 & x \geq -2 \\ -x - 2 & x < -2 \end{cases} $$ - So for $x \geq -2$, the graph is the line $y = x + 2$. - For $x < -2$, the graph is the line $y = -x - 2$. 6. The vertex of the graph is at $x = -2$, where $f(-2) = 0$. 7. Summary: - a) Graph is $y=0$ for $x<0$ and $y=2x$ for $x \geq 0$. - b) Graph is $y = -x - 2$ for $x < -2$ and $y = x + 2$ for $x \geq -2$. These piecewise linear graphs can be sketched accordingly.