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To determine which of the following functions has the largest value when [tex]\( x = 5 \)[/tex]:
[tex]\[ \begin{align*} p(x) &= 2x^2 + 4x + 11 \\ h(x) &= 4^x \\ s(x) &= 10x \\ \end{align*} \][/tex]
Step-by-step evaluation of each function at [tex]\( x = 5 \)[/tex]:
1. Calculate [tex]\( p(5) \)[/tex]:
[tex]\[ p(5) = 2(5)^2 + 4(5) + 11 \][/tex]
First, calculate [tex]\( 2(5)^2 \)[/tex]:
[tex]\[ 2(5)^2 = 2(25) = 50 \][/tex]
Next, calculate [tex]\( 4(5) \)[/tex]:
[tex]\[ 4(5) = 20 \][/tex]
Add these results together along with the constant 11:
[tex]\[ p(5) = 50 + 20 + 11 = 81 \][/tex]
2. Calculate [tex]\( h(5) \)[/tex]:
[tex]\[ h(5) = 4^5 \][/tex]
Calculate [tex]\( 4^5 \)[/tex] through exponentiation:
[tex]\[ 4^5 = 1024 \][/tex]
3. Calculate [tex]\( s(5) \)[/tex]:
[tex]\[ s(5) = 10(5) \][/tex]
Calculate [tex]\( 10(5) \)[/tex]:
[tex]\[ 10(5) = 50 \][/tex]
Summary of the values calculated at [tex]\( x = 5 \)[/tex]:
[tex]\[ \begin{align*} p(5) &= 81 \\ h(5) &= 1024 \\ s(5) &= 50 \\ \end{align*} \][/tex]
To find the largest value, compare these results:
[tex]\[ p(5) = 81, \quad h(5) = 1024, \quad s(5) = 50 \][/tex]
Among the values [tex]\( 81 \)[/tex], [tex]\( 1024 \)[/tex], and [tex]\( 50 \)[/tex], the largest value is [tex]\( 1024 \)[/tex].
Thus, the function with the largest value at [tex]\( x = 5 \)[/tex] is [tex]\( h(x) = 4^x \)[/tex].
[tex]\[ \begin{align*} p(x) &= 2x^2 + 4x + 11 \\ h(x) &= 4^x \\ s(x) &= 10x \\ \end{align*} \][/tex]
Step-by-step evaluation of each function at [tex]\( x = 5 \)[/tex]:
1. Calculate [tex]\( p(5) \)[/tex]:
[tex]\[ p(5) = 2(5)^2 + 4(5) + 11 \][/tex]
First, calculate [tex]\( 2(5)^2 \)[/tex]:
[tex]\[ 2(5)^2 = 2(25) = 50 \][/tex]
Next, calculate [tex]\( 4(5) \)[/tex]:
[tex]\[ 4(5) = 20 \][/tex]
Add these results together along with the constant 11:
[tex]\[ p(5) = 50 + 20 + 11 = 81 \][/tex]
2. Calculate [tex]\( h(5) \)[/tex]:
[tex]\[ h(5) = 4^5 \][/tex]
Calculate [tex]\( 4^5 \)[/tex] through exponentiation:
[tex]\[ 4^5 = 1024 \][/tex]
3. Calculate [tex]\( s(5) \)[/tex]:
[tex]\[ s(5) = 10(5) \][/tex]
Calculate [tex]\( 10(5) \)[/tex]:
[tex]\[ 10(5) = 50 \][/tex]
Summary of the values calculated at [tex]\( x = 5 \)[/tex]:
[tex]\[ \begin{align*} p(5) &= 81 \\ h(5) &= 1024 \\ s(5) &= 50 \\ \end{align*} \][/tex]
To find the largest value, compare these results:
[tex]\[ p(5) = 81, \quad h(5) = 1024, \quad s(5) = 50 \][/tex]
Among the values [tex]\( 81 \)[/tex], [tex]\( 1024 \)[/tex], and [tex]\( 50 \)[/tex], the largest value is [tex]\( 1024 \)[/tex].
Thus, the function with the largest value at [tex]\( x = 5 \)[/tex] is [tex]\( h(x) = 4^x \)[/tex].
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