Answer :
Sure! Let's compare the two given functions [tex]\( y = 3^{-x} \)[/tex] and [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] step by step.
1. Rewrite the functions:
- The first function is given as [tex]\( y = 3^{-x} \)[/tex].
- The second function is given as [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex].
2. Understand the transformations:
- For [tex]\( y = 3^{-x} \)[/tex], the exponent is [tex]\(-x\)[/tex]. This means you take the reciprocal of the base [tex]\(3\)[/tex] and then raise it to the power [tex]\(x\)[/tex], i.e., [tex]\( 3^{-x} = \left(\frac{1}{3}\right)^x \)[/tex].
3. Simplify and compare the forms:
- Both functions actually translate to the same mathematical expression.
- Specifically, [tex]\( 3^{-x} \)[/tex] and [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] mean the same thing.
4. Analyze the graphs:
- Since both expressions are identical, they represent the same relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Therefore, the graphs of [tex]\( y = 3^{-x} \)[/tex] and [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] will be indistinguishable from one another.
5. Conclusion:
- The graphs of [tex]\( y = 3^{-x} \)[/tex] and [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] are identical. They coincide perfectly on the coordinate plane.
Thus, the correct answer is:
The graphs are the same.
1. Rewrite the functions:
- The first function is given as [tex]\( y = 3^{-x} \)[/tex].
- The second function is given as [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex].
2. Understand the transformations:
- For [tex]\( y = 3^{-x} \)[/tex], the exponent is [tex]\(-x\)[/tex]. This means you take the reciprocal of the base [tex]\(3\)[/tex] and then raise it to the power [tex]\(x\)[/tex], i.e., [tex]\( 3^{-x} = \left(\frac{1}{3}\right)^x \)[/tex].
3. Simplify and compare the forms:
- Both functions actually translate to the same mathematical expression.
- Specifically, [tex]\( 3^{-x} \)[/tex] and [tex]\( \left(\frac{1}{3}\right)^x \)[/tex] mean the same thing.
4. Analyze the graphs:
- Since both expressions are identical, they represent the same relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Therefore, the graphs of [tex]\( y = 3^{-x} \)[/tex] and [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] will be indistinguishable from one another.
5. Conclusion:
- The graphs of [tex]\( y = 3^{-x} \)[/tex] and [tex]\( y = \left(\frac{1}{3}\right)^x \)[/tex] are identical. They coincide perfectly on the coordinate plane.
Thus, the correct answer is:
The graphs are the same.
