Problem 55 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. \(h(x)=-2 x-5 .\) Find \(x\) so that \(h(x)=0\)

Step-by-Step Solution

Answer

The value of x that makes h(x) equal to 0 is \(x = -\frac{5}{2}\).

Step 1: Write down the function h(x)

We are given the function h(x) as: \(h(x) = -2x - 5\).

Step 2: Set h(x) equal to 0

To find the value of x that makes h(x) equal to 0, we set the function equal to 0 and solve for x: \[-2x - 5 = 0\]

Step 3: Solve for x

Move '5' to the other side of the equation: \[-2x = 5\] Now, divide by -2 to get the value of x: \[x = \frac{5}{-2}\] \[x = -\frac{5}{2}\] So, the value of x that makes h(x) equal to 0 is \(x = -\frac{5}{2}\).

Key Concepts

Understanding Functions

Functions are like machines that take an input and give an output. Think of them as rules or equations that define how one thing depends on another. For example, in the function \(h(x) = -2x - 5\), \(x\) is the input. Whatever value you put in, the function does something to it, and gives an output.

• The equation tells you exactly what to do with \(x\). Here, you multiply \(x\) by \(-2\) and then subtract \(5\).
• Understanding functions is crucial because they describe relationships in math and many real-world scenarios, like calculating interest rates or determining travel time.

Functions can be linear like \(h(x)\), or more complex like \(g(x) = x^2 + 7x + 2\), where the output depends on higher powers of the input.

Finding Zeros of Functions

Finding the zeros of a function means finding the value(s) of \(x\) that make the function equal to zero. This is an important skill because it helps you solve equations and discover interesting properties of functions. In our example, we found the zero of the function \(h(x) = -2x - 5\). To find the zero, we set the function equal to zero: \[-2x - 5 = 0\]Then, we solve for \(x\) by:

• Moving the constant term \(-5\) across the equation (changing its sign): \(-2x = 5\)
• Next, dividing both sides by \(-2\) to solve for \(x\): \(x = \frac{5}{-2}\)

The solution tells us that when \(x = -\frac{5}{2}\), \(h(x)\) becomes zero. This is essential for understanding when and how functions will intersect the x-axis.

Simplifying Expressions

Simplifying expressions is about reducing a mathematical expression into its simplest form where it's easier to work with or understand. In the context of linear functions, like \(h(x)\), simplifying might involve combining like terms or solving for a variable.For instance, whenever you move a term from one side of an equation to the other, you are simplifying by changing the sign and making the equation easier. In the equation \(-2x - 5 = 0\), simplifying involves moving \(-5\) to the other side and changing it to \(5\), which helps isolate \(x\):

• Combining terms: When solving, quickly see if something should be added or subtracted to organize the equation better.
• Maintaining balance: Always perform the same operation on both sides, like dividing by \(-2\) to keep the equation balanced.

Simplifying is fundamental in solving equations because it makes the operation straightforward and leads you directly to the solution.