Question

(a) $y = 5x$
$\bigcirc$ $f$
$\bigcirc$ $f$
$\bigcirc$ $g$
$\bigcirc$ $g$

(b) $y = 5^x$
$\bigcirc$ $f$
$\bigcirc$ $f$
$\bigcirc$ $g$
$\bigcirc$ $g$

(c) $y = x^5$
$\bigcirc$ $f$
$\bigcirc$ $f$
$\bigcirc$ $g$
$\bigcirc$ $g$

(d) $y = \sqrt5{x}$
$\bigcirc$ $f$
$\bigcirc$ $f$
$\bigcirc$ $g$
$\bigcirc$ $g$

Explanation:

Step1: Analyze linear function $y=5x$

This is a straight line with slope 5, passing through the origin. It matches line $G$, which is the steep straight line through $(0,0)$.

Step2: Analyze exponential function $y=5^x$

This is an exponential growth function: it approaches 0 as $x\to-\infty$, passes through $(0,1)$, and increases rapidly for $x>0$. It matches curve $f$.

Step3: Analyze power function $y=x^5$

This is an odd power function: it passes through $(0,0)$, is negative for $x<0$, positive for $x>0$, and has a steeper curve for large $|x|$. It matches curve $F$ (the red curve).

Step4: Analyze root function $y=\sqrt[5]{x}=x^{\frac{1}{5}}$

This is the inverse of $y=x^5$: it is odd, passes through $(0,0)$, increases slowly, and is defined for all real $x$. It matches curve $g$ (the lower blue curve).

Answer:

(a) G
(b) f
(c) F
(d) g