Question: User: g(x)=3^{x-2}\nh(x)=2\cdot 3^x\ni(x)=2^{x-3}-2\ntop-left: g(x)=3^{x-2}\ncenter: h(x)=2\cdot 3^x\nbottom-left: i(x)=2^{x-3}-2
1. The problem gives three functions: $$g(x)=3^{x-2}$$, $$h(x)=2\cdot 3^x$$, and $$i(x)=2^{x-3}-2$$. We are asked to understand or analyze these exponential functions.
2. Exponential functions have the form $$a^{x+c}$$ or $$k\cdot a^x + d$$ where $$a$$ is the base, $$x$$ is the variable, and $$c, k, d$$ are constants.
3. For $$g(x)=3^{x-2}$$, the base is 3 and the exponent is shifted by -2, meaning the graph shifts right by 2 units.
4. For $$h(x)=2\cdot 3^x$$, the base is 3, multiplied by 2, which vertically stretches the graph by a factor of 2.
5. For $$i(x)=2^{x-3}-2$$, the base is 2, the exponent is shifted by -3 (right shift by 3), and the whole function is shifted down by 2 units.
6. These transformations affect the shape and position of the graphs but keep the exponential growth or decay behavior.
Final answer: The functions are exponential with bases 3 or 2, shifted and scaled as described above.