Evaluate the sum of a function

In summary, evaluating the sum of a function allows us to determine the total value of the function at a specific input or set of inputs, providing insight into its behavior and allowing for predictions about its values at other inputs. This is done by substituting the input values into the function and performing mathematical operations. There are various methods for evaluating the sum of a function, such as substitution, factoring, and using function properties. It is not possible to evaluate the sum of a function without knowing its equation. Evaluating the sum of a function differs from evaluating the value at a specific point, as the former provides an overall understanding of the function's behavior while the latter gives a specific data point on its graph.
  • #1
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 $h(x)=\dfrac{9^x}{9^x+3}$.
 
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  • #2
anemone said:
Evaluate $h\left( \dfrac{1}{401} \right)+h\left( \dfrac{2}{401} \right)+\cdots+h\left( \dfrac{400}{401} \right)$ if $h(x)=\dfrac{9^x}{9^x+3}$.

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.
 
  • #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.

Well done and thanks for participating, Pranav! I think this problem is doable only if one recognizes that in this case, $h(x)+h(1-x)=1$!(Happy)
 

FAQ: Evaluate the sum of a function

What is the purpose of evaluating the sum of a function?

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.

How do you evaluate the sum of a function?

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.

What are the common methods for evaluating the sum of a function?

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.

Can you evaluate the sum of a function without knowing the function's equation?

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.

How does evaluating the sum of a function differ from evaluating the value of a function at a specific point?

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.

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