Question

given f(x)=3^x-2 and g(x)=f(3x)+4, write the function rule for function g and describe the types of transformations that occur between function f and function g.

Answer 1

Answer:

$g(x)=3^{3x}+2$

The transformation that occurs from $f(x)\rightarrow g(x)$ is compression by 3 units horizontally and shift  of 4 units upwards.

Step-by-step explanation:

Given

$f(x)=3^x-2$

$g(x)=f(3x)+4$

Translation Rules:

$f(x)\rightarrow f(cx)$

If $c>1$ the function is compressed $c$ units horizontally.

If $c$ and $c>0$  the function stretches $c$ units horizontally.

$f(x)\rightarrow f(x)+c$

If $c>0$ the function shifts $c$ units to the up.

If $c$ the function shifts $c$ units to the down.

Applying the rules to $f(x)$

Step 1

$f(x)\rightarrow f(3x)$

$f(3x)=3^{3x}-2$

[compressed $3$ units horizontally]

Step 2

$f(3x)\rightarrow f(3x)+4=g(x)$

$f(3x)+4=3^{3x}-2+4$

∴ $g(x)=3^{3x}+2$

[shifts 4 nits up]

∴ The transformation that occurs from $f(x)\rightarrow g(x)$ is compression by 3 units horizontally and shift  of 4 units upwards.