When we want to compare how quickly different types of functions grow as we increase \(x\) (the input) toward infinity, we generally talk about exponential functions and quadratic functions.
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Understand the Types of Growth:
- Quadratic Functions: These functions are of the form \(f(x) = ax^2 + bx + c\). They grow faster than linear functions but slower than exponential functions. As \(x\) gets very large, the \(x^2\) part makes the function rise, but it does so in a way that eventually gets outpaced by exponential functions.
- Exponential Functions: These are of the form \(g(x) = a \cdot b^x\) (where \(b > 1\)). Exponential functions grow much faster than quadratic functions because they involve raising a constant to the power of \(x\). As \(x\) increases, the value of \(b^x\) increases very quickly.
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Arranging the Functions:
- As \(x\) approaches infinity, exponential functions will outgrow quadratic functions significantly. In simple terms, if you imagine plotting these functions on a graph, the exponential function will shoot up much higher than the quadratic function as \(x\) gets really large.
So, if you are arranging these types of functions in order from the fastest growth to the slowest growth as \(x\) increases toward infinity, you would arrange them like this:
- Fastest Growth: Exponential functions (like \(g(x) = 2^x\))
- Slower Growth: Quadratic functions (like \(f(x) = x^2\))
In summary, exponential functions grow faster than quadratic functions as \(x\) becomes very large (approaching infinity).