Answer: [tex]The\text{ correct function is f\lparen x\rparen= 2}^{x-1}\text{ \lparen option B\rparen}[/tex]Explanation:
Given:
An exponential graph with a given point (0, 0.5)
To find:
the function that corresponds to the graph
To determine the correct function, we will substitute x in each of the functions with the x coordinate of the given point and compare the result with the y coordinate of the given point.
[tex]\begin{gathered} for\text{ point: \lparen0, 0.5\rparen: } \\ x\text{ = 0, y = f\lparen x\rparen = 0.5} \\ \\ a)\text{ f\lparen x\rparen = 2}\times2^{x-1} \\ when\text{ x = 0} \\ f(x)\text{ = 2}\times2^{0-1}\text{ = 2}\times2^{-1} \\ f(x)\text{ = 2}\times\frac{1}{2}\text{ = 1} \\ This\text{ is different from the f\lparen x\rparen value of the given point} \end{gathered}[/tex][tex]\begin{gathered} b)\text{ f\lparen x\rparen= 2}^{x-1} \\ f(x)\text{ = 2}^{0-1}\text{ = 2}^{-1} \\ f(x)\text{ = }\frac{1}{2}\text{ = 0.5} \\ This\text{ is the same as the f\lparen x\rparen value of the given point} \end{gathered}[/tex][tex]\begin{gathered} c)\text{ f\lparen x\rparen = 2}\times2^x \\ f(x)\text{ = 2}\times2^0\text{ = 2}\times1 \\ f(x)\text{ = 2} \\ This\text{ is different from the f\lparen x\rparen value of the given point} \end{gathered}[/tex][tex]\begin{gathered} d)\text{ f\lparen x\rparen= 2}^x \\ f(x)\text{ = 2}^0\text{ = 1} \\ This\text{ is different from the f\lparen x\rparen value of the given point} \end{gathered}[/tex][tex]\begin{gathered} e)\text{ f\lparen x\rparen= 2}^{x+1} \\ f(x)\text{ = 2}^{0+1}\text{ = 2}^1 \\ f(x)\text{ = 2} \\ This\text{ is different from the f\lparen x\rparen value of the given point} \end{gathered}[/tex]
Hence, the correct function that represents the graph is f(x) = 2^(x-1)
