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Chapter 5: Problem 27

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. \((g-h)(-3)\)

Short Answer

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Step by step solution

01

Determine the functions needed

Identify the functions given: \( g(x) = 2x \)\( h(x) = x - 3 \)
02

Define the combined function

Define the combined function \((g - h)(x)\). This is calculated by subtracting \(h(x)\) from \(g(x)\).\[(g - h)(x) = g(x) - h(x)\]\[(g - h)(x) = 2x - (x - 3)\]
03

Simplify the expression

Simplify the combined function expression:\[(g - h)(x) = 2x - x + 3\]\[(g - h)(x) = x + 3\]
04

Evaluate the combined function at \(x = -3\)

Substitute \(x = -3\) into the simplified combined function:\[(g - h)(-3) = (-3) + 3\]\[(g - h)(-3) = 0\]

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Key Concepts

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

Combining Functions
In mathematics, combining functions allows us to perform operations between two or more functions. This exercise involves the functions \(g(x) = 2x\) and \(h(x) = x - 3\). Let鈥檚 talk about how combining functions works.

When we combine functions, we can add, subtract, multiply, or divide them. In our example, we are subtracting function \(h(x)\) from function \(g(x)\).

To form the combined function \((g - h)(x)\), we subtract \(h(x)\) from \(g(x)\):
  • Write out each function: \(g(x) = 2x\) and \(h(x) = x - 3\)
  • Subtract \(h(x)\) from \(g(x)\): \[(g - h)(x) = g(x) - h(x)\]
  • Substitute the given functions into the equation: \[(g - h)(x) = 2x - (x - 3)\]
It鈥檚 important to carefully handle the subtraction to avoid mistakes in combining the functions.
Evaluating Functions
Evaluating functions means finding the value of a function for a specific input. After combining the functions, we get \((g - h)(x) = x + 3\). Now, we need to evaluate this combined function at a specific value of \(x\), which in our case is \(-3\).

To evaluate \((g - h)(x)\) at \(x = -3\):
  • First, simplify the combined function: \( (g - h)(x) = x + 3 \)
  • Next, substitute \(x = -3\) into the expression: \[(g - h)(-3) = -3 + 3\]
  • Finally, calculate the value: \[(g - h)(-3) = 0\]
Therefore, the value of \((g - h)(-3)\) is \(0\). Evaluating functions is a crucial step as it helps us understand their behavior at specific points.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This helps in making complex expressions easier to understand and work with. When we were combining the functions \(g(x)\) and \(h(x)\), we reached a point where we had: \[(g - h)(x) = 2x - (x - 3)\]

Here are the steps to simplify this expression further:
  • Distribute the negative sign across \(h(x)\): \[(g - h)(x) = 2x - x + 3\]
  • Combine like terms: \[2x - x = x\]
  • So, the simplified form of the combined function is: \[(g - h)(x) = x + 3\]
Simplifying the expression makes it much easier to subsequently evaluate it at any given value of \(x\). Simplification is a fundamental skill in algebra that helps in solving equations more efficiently.

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