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anemone
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Evaluate $h\left( \dfrac{1}{401} \right)+h\left( \dfrac{2}{401} \right)+\cdots+h\left( \dfrac{400}{401} \right)$ if $h(x)=\dfrac{9^x}{9^x+3}$.
anemone said:Evaluate $h\left( \dfrac{1}{401} \right)+h\left( \dfrac{2}{401} \right)+\cdots+h\left( \dfrac{400}{401} \right)$ if $h(x)=\dfrac{9^x}{9^x+3}$.
Pranav said:Notice that h(x)+h(1-x)=1.
Hence,
$$h\left(\frac{1}{401}\right)+h\left(\frac{400}{401}\right)=1$$
$$h\left(\frac{2}{401}\right)+h\left(\frac{399}{401}\right)=1$$
$$.$$
$$.$$
$$.$$
$$h\left(\frac{200}{401}\right)+h\left(\frac{201}{401}\right)=1$$
So the sum is 200.
The purpose of evaluating the sum of a function is to find the total value of the function at a specific input or set of inputs. This can help us understand the behavior of the function and make predictions about its values at other inputs.
To evaluate the sum of a function, you need to substitute the input values into the function and then perform the necessary mathematical operations. For example, if the function is f(x) = 2x + 3 and you want to evaluate it at x = 5, you would substitute 5 for x and get f(5) = 2(5) + 3 = 13.
There are several common methods for evaluating the sum of a function, including substitution, factoring, and using the properties of functions (such as the distributive property or the sum/difference identities). The method used will depend on the specific function and input values.
No, in order to evaluate the sum of a function, you need to know the function's equation. Without the equation, you do not have a way to determine the output values for a given set of inputs.
Evaluating the sum of a function involves finding the total value of the function for a specific set of inputs, while evaluating the value of a function at a specific point involves finding the value of the function at a single input. In other words, evaluating the sum of a function gives you the overall picture of the function's behavior, while evaluating the value at a specific point gives you a specific data point on the function's graph.