Math Problem Statement

(a) For the linear function ( f(x)=x+5 ), evaluate ( f(-2) ) and ( f(4) ).

(b) Graph ( f ). How can the graph of ( f ) be used to determine the zero of ( f ) ?

(c) Find the zero of ( f ).

(a) ( f(-2)= ) ( \square )

(Type an integer or a fraction.)

Solution

Let's work through the problem step by step.

Part (a): Evaluate f(2)f(-2) and f(4)f(4).

The linear function is given by f(x)=x+5f(x) = x + 5.

  1. Evaluate f(2)f(-2): f(2)=(2)+5=3f(-2) = (-2) + 5 = 3

  2. Evaluate f(4)f(4): f(4)=4+5=9f(4) = 4 + 5 = 9

So:

  • f(2)=3f(-2) = 3
  • f(4)=9f(4) = 9

Part (b): Graph f(x)=x+5f(x) = x + 5

The graph of f(x)=x+5f(x) = x + 5 is a straight line with a slope of 1 and a y-intercept at (0,5)(0, 5).

Using the Graph to Determine the Zero of ff:

The zero of the function f(x)f(x) is the value of xx where f(x)=0f(x) = 0. On the graph, this is the point where the line crosses the x-axis.

Part (c): Find the Zero of f(x)=x+5f(x) = x + 5

To find the zero, set f(x)=0f(x) = 0 and solve for xx:

x+5=0x + 5 = 0 x=5x = -5

So, the zero of the function is x=5x = -5.

Summary:

  • (a) f(2)=3f(-2) = 3
  • (b) The graph of f(x)=x+5f(x) = x + 5 crosses the x-axis at x=5x = -5.
  • (c) The zero of f(x)f(x) is x=5x = -5.

Would you like further details on any part of this solution?


Here are five related questions you might explore:

  1. What is the slope of the function f(x)=x+5f(x) = x + 5?
  2. How does changing the value of the constant term (e.g., x+3x + 3 instead of x+5x + 5) affect the graph?
  3. What is the y-intercept of the function f(x)=x+5f(x) = x + 5?
  4. How would you find the zero of the function f(x)=2x+7f(x) = 2x + 7?
  5. What does the zero of a function represent in a real-world context?

Tip: The slope of a linear function tells you how steep the line is, while the y-intercept indicates where the line crosses the y-axis.

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Math Problem Analysis

Mathematical Concepts

Linear Functions
Graphing
Zero of a Function

Formulas

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Theorems

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Suitable Grade Level

Grades 7-9