Answer :
To determine which of the given functions represent exponential growth, let's analyze them one by one.
1. Function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x + 2 \][/tex]
This is a linear function, as it represents a straight-line equation with a slope of 1 and a y-intercept of 2. Linear functions are not exponential growth.
2. Function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x \][/tex]
This function is of the form [tex]\( a \cdot b^x \)[/tex], where [tex]\( a = 1 \)[/tex] and [tex]\( b = 2 \)[/tex]. For exponential growth, [tex]\( b \)[/tex] needs to be greater than 1. Here, [tex]\( b \)[/tex] is 2, which meets the criteria for exponential growth.
3. Function [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = 3x \][/tex]
This is another linear function, where the slope is 3. Like [tex]\( f(x) \)[/tex], linear functions do not represent exponential growth.
4. Function [tex]\( k(x) \)[/tex]:
[tex]\[ k(x) = x^2 \][/tex]
This is a quadratic function because it has the variable [tex]\( x \)[/tex] raised to the power of 2. Quadratic functions describe parabolic shapes and are not exponential growth functions.
Based on our analysis, the function that represents exponential growth is:
[tex]\[ y = g(x) = 2^x \][/tex]
1. Function [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x + 2 \][/tex]
This is a linear function, as it represents a straight-line equation with a slope of 1 and a y-intercept of 2. Linear functions are not exponential growth.
2. Function [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = 2^x \][/tex]
This function is of the form [tex]\( a \cdot b^x \)[/tex], where [tex]\( a = 1 \)[/tex] and [tex]\( b = 2 \)[/tex]. For exponential growth, [tex]\( b \)[/tex] needs to be greater than 1. Here, [tex]\( b \)[/tex] is 2, which meets the criteria for exponential growth.
3. Function [tex]\( h(x) \)[/tex]:
[tex]\[ h(x) = 3x \][/tex]
This is another linear function, where the slope is 3. Like [tex]\( f(x) \)[/tex], linear functions do not represent exponential growth.
4. Function [tex]\( k(x) \)[/tex]:
[tex]\[ k(x) = x^2 \][/tex]
This is a quadratic function because it has the variable [tex]\( x \)[/tex] raised to the power of 2. Quadratic functions describe parabolic shapes and are not exponential growth functions.
Based on our analysis, the function that represents exponential growth is:
[tex]\[ y = g(x) = 2^x \][/tex]
