Problem 74
Question
Work each problem. If the point \((-3,2)\) lies on the graph of \(f,\) then \(f(\quad)=\text {_____}\)
Step-by-Step Solution
Verified Answer
f(-3) = 2.
1Step 1: Identify the Given Point
The point given is
(-3, 2).
This implies that when the input to the function is -3, the output is 2.
2Step 2: Formulate the Function Output
Interpret the meaning of the point. Since
(-3, 2)
is on the graph of
f,
it means
f(-3) = 2.
3Step 3: Complete the Function Equation
Fill in the blank:
f(
-3
) = 2.
Therefore,
f(
-3
) = 2.
Key Concepts
Graph InterpretationInput-Output RelationshipOrdered Pairs
Graph Interpretation
Interpreting a graph is like reading a story that is told in visual form. In mathematics, graphs of functions tell us how the input and output are related. Imagine the x-axis as the inputs and the y-axis as the outputs. Each point on the graph represents this relationship.
For instance, the point \((-3, 2)\) suggests that the input \(-3\) results in the output \(2\) for the function \(f\). When looking at a graph, these points give concrete insight into what the function does. You can think of it as a lookup table where each input matches to its corresponding output, as given by the y-value of the point.
For instance, the point \((-3, 2)\) suggests that the input \(-3\) results in the output \(2\) for the function \(f\). When looking at a graph, these points give concrete insight into what the function does. You can think of it as a lookup table where each input matches to its corresponding output, as given by the y-value of the point.
- X-axis: The horizontal line where inputs lie.
- Y-axis: The vertical line where outputs are shown.
- Graph Point: Shows how specific inputs are related to outputs.
Input-Output Relationship
The input-output relationship describes how each input value affects the output in a function. This is a core concept of functions in general. In the function notation \(f(x)\), \(x\) is the input, and \(f(x)\) is the output.
So, when we look at a point like \((-3, 2)\) on a function's graph, it tells us that an input of \(-3\) produces an output of \(2\). You can liken this relationship to a machine: you put an input into the machine, and it processes it to give you an output.
This understanding is crucial not only for plotting graphs but also for solving function-based problems. By knowing how inputs turn into outputs, one can predict the outcomes, even without seeing the graph.
So, when we look at a point like \((-3, 2)\) on a function's graph, it tells us that an input of \(-3\) produces an output of \(2\). You can liken this relationship to a machine: you put an input into the machine, and it processes it to give you an output.
This understanding is crucial not only for plotting graphs but also for solving function-based problems. By knowing how inputs turn into outputs, one can predict the outcomes, even without seeing the graph.
- Function notation: Expresses the input-output process.
- Input: What you "put into" the function.
- Output: The result of the function's operation on the input.
Ordered Pairs
Ordered pairs are foundational in graphing and function analysis. An ordered pair is comprised of two elements, typically presented as \((x, y)\), that indicate a point on the graph.
The first element \(x\) is the input, and the second element \(y\) is the output. In the example of \((-3, 2)\), the input of the function is \(-3\) and the output is \(2\).
By using ordered pairs, we can plot the behavior of the function across a series of inputs and outputs, connecting the dots to reveal the function's graph.
The first element \(x\) is the input, and the second element \(y\) is the output. In the example of \((-3, 2)\), the input of the function is \(-3\) and the output is \(2\).
By using ordered pairs, we can plot the behavior of the function across a series of inputs and outputs, connecting the dots to reveal the function's graph.
- X-component: Represents the position along the x-axis.
- Y-component: Represents the position along the y-axis.
- Connection: Ordered pairs illustrate the input-output drag of function operations.
Other exercises in this chapter
Problem 73
Solve each formula for the specified variable.} \(s=\frac{1}{2} g t^{2}\) for \(g\) (Distance traveled by a falling object)
View solution Problem 73
Approximate each expression to the nearest hundredth. $$\text { 73. } \frac{\sqrt{\pi-1}}{\sqrt{1+\pi}}$$
View solution Problem 74
Solve each formula for the specified variable.} \(A=\frac{24 f}{B(p+1)}\) for \(p\) (Approximate annual interest rate)
View solution Problem 74
Approximate each expression to the nearest hundredth. $$\sqrt[3]{4.5 \times 10^{5}+3.7 \times 10^{2}}$$
View solution