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Chapter 2: Problem 8

Let \(f(x)=-2 x+3\) and \(g(x)=x^{2}-5 .\) Find each of the following. $$ f(4)+g(4) $$

Short Answer

Expert verified
6

Step by step solution

01

- Evaluate f(x) at x=4

Substitute x = 4 into the function f(x). So, it becomes: \[ f(4) = -2(4) + 3 \] Calculate the result: \[ f(4) = -8 + 3 = -5 \]
02

- Evaluate g(x) at x=4

Substitute x = 4 into the function g(x). So, it becomes: \[ g(4) = (4)^2 - 5 \] Calculate the result: \[ g(4) = 16 - 5 = 11 \]
03

- Add f(4) and g(4)

Now, add the results from Step 1 and Step 2: \[ f(4) + g(4) = -5 + 11 = 6 \]

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Key Concepts

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

Linear Functions
A linear function is a type of function that creates a straight line when graphed. The general form of a linear function is y = mx + b, where:
  • m is the slope
  • b is the y-intercept
In the exercise example, the function f(x) = -2x + 3 is a linear function. Here, the slope m is -2 and the y-intercept b is 3. The slope tells you how steep the graph is, indicating that for each unit increase in x, the value of y decreases by 2. The y-intercept is the point where the line crosses the y-axis. In this case, it's 3. To calculate the value of f(x) at x = 4, we substitute 4 for x:f(4) = -2(4) + 3 = -8 + 3 = -5.This step-by-step method helps in evaluating any linear function at a given point.
Quadratic Functions
A quadratic function is a function that creates a parabola (a curve) when graphed. Its general form is y = ax^2 + bx + c. In the given exercise, g(x) = x^2 - 5 is a quadratic function where a is 1, and both b and c are 0 and -5 respectively. Quadratic functions have a variable raised to the second power, making them different from linear functions that are first degree. To evaluate g(x) at x = 4, substitute the value 4 for x: g(4) = (4)^2 - 5 = 16 - 5 = 11.This shows how the output of a quadratic function is calculated and leads to proper understanding step by step.
Function Operations
Function operations allow you to combine functions in various ways, such as addition, subtraction, multiplication, or division. These operations follow specific rules. In the exercise, we are asked to add two functions f(x) and g(x) after evaluating each at x = 4. First, evaluate both functions separately:
  • f(4) = -5 (from the evaluated form)
  • g(4) = 11 (also from evaluated form)
Then, we add the results of f(4) and g(4): f(4) + g(4) = -5 + 11 = 6.Adding functions is straightforward; you just combine their outputs at the same input value. By understanding how to perform these operations, you can solve more complex problems involving multiple functions easily.

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Most popular questions from this chapter

Solve using a graphing calculator. Weekly pay at The Furniture Gallery is \(\$ 450\) plus a \(3.5 \%\) sales commission. If a salesperson's pay was \(\$ 601.03,\) what did that salesperson's sales total?

Let \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) be two distinct points on the graph of \(y=m x+b .\) Use the fact that both pairs are solutions of the equation to prove that \(m\) is the slope of the line given by \(y=m x+b\) (Hint: Use the slope formula.)

Determine whether each equation is linear. Find the slope of any nonvertical lines. $$ 3 y=7(2 x-4) $$

Use a graph to estimate the solution in each of the following. Be sure to use graph paper and a straightedge. Cal's Parking charges \(\$ 5.00\) to park plus \(50 \mathrm{c}\) for each \(15-\mathrm{min}\) unit of time. Estimate how long someone can park for \(\$ 9.50 .\)

Find the indicated function values. $$ f(x)=\left\\{\begin{array}{ll}{x,} & {\text { if } x<0} \\ {2 x+1,} & {\text { if } x \geq 0}\end{array}\right. $$ a) \(f(-5) \quad\) b) \(f(0)\) c) \(f(10)\)
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