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Chapter 0: Problem 48

Is the point \((-2,12)\) on the graph of the function \(f(x)=x(5+x)(4-x) ?\)

Short Answer

Expert verified
No, the point \((-2, 12)\) is not on the graph of the function.

Step by step solution

01

Identify the function and the point

The function given is \(f(x) = x(5+x)(4-x)\). The point we need to check is \((-2, 12)\). To determine if the point lies on the graph, substitute \(x = -2\) into the function and see if the resulting \(f(x)\) equals 12.
02

Substitute the x-value into the function

Substitute \(x = -2\) into the function: \[ f(-2) = (-2)(5 + (-2))(4 - (-2)) \]
03

Simplify the expression

Simplify the expression step by step: \[ f(-2) = (-2)(5 - 2)(4 + 2) \] \[ f(-2) = (-2)(3)(6) \] \[ f(-2) = -36 \]
04

Compare the resulting value with the y-value of the point

We have \(f(-2) = -36\). Compare this with the y-coordinate of the point \( ( -2, 12 ) \). Since \(-36 eq 12\), the point is not on the graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
When we talk about the substitution method in the context of this problem, it means evaluating the function by replacing the variable with a specific value. In this case, we substitute the x-value from the point \((-2, 12)\) into the function \(f(x) = x(5+x)(4-x)\). This helps us to check if the y-value from the point matches the function's output at that x-value. It's a straightforward technique to verify points on a function graph.
point on a graph
A point like \((-2, 12)\) on a graph represents a specific x-value and its corresponding y-value. To verify if this point lies on the graph of a given function, we follow these steps:

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