Question

how does the graph of $f(x) = \sqrt3{x}$ change when $f(x)$ is replaced by $f(x + c)$ and $c$ is positive? (1 point) \
\
\bigcirc the graph shifts down. \
\
\bigcirc the graph shifts left. \
\
\bigcirc the graph shifts to the right. \
\
\bigcirc the graph shifts up.

Explanation:

To determine the graph transformation, we use the rule for horizontal shifts of functions. For a function \( y = f(x) \), the transformation \( y = f(x + c) \) (where \( c>0 \)) represents a horizontal shift. The rule is: if we have \( f(x + c) \) with \( c>0 \), the graph of \( f(x) \) shifts left by \( c \) units. This is because to get the same output as \( f(x) \) at \( x \), we now need \( x + c \) to be equal to the original input, so \( x \) must be smaller (shift left) to compensate for the added \( c \). Applying this to \( f(x)=\sqrt[3]{x} \) and \( f(x + c)=\sqrt[3]{x + c} \) with \( c>0 \), the graph shifts left.

Answer:

B. The graph shifts left.