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Chapter 10: Problem 1

For each pair of functions, determine \(h(x)=f(x)+g(x)\) a) \(f(x)=|x-3|\) and \(g(x)=4\) b) \(f(x)=3 x-5\) and \(g(x)=-x+2\) c) \(f(x)=x^{2}+2 x\) and \(g(x)=x^{2}+x+2\) d) \(f(x)=-x-5\) and \(g(x)=(x+3)^{2}\)

Short Answer

Expert verified
(a) \(|x-3|+4\), (b) \(2x-3\), (c) \(2x^2+3x+2\), (d) \(x^2 + 5x + 4\).

Step by step solution

01

Title - Understanding function addition

When combining two functions, the new function is formed by adding the outputs of the two functions for the same input. Mathematically, this is represented as \(h(x) = f(x) + g(x)\).
02

Title - Adding the functions for each part

Calculate \(h(x)\) for each given pair of functions.
03

Part (a) - Calculate \(h(x)\) when \(f(x)=|x-3|\) and \(g(x)=4\)

Given: \(f(x) = |x-3|\) and \(g(x) = 4\). Then, \(h(x) = f(x) + g(x) = |x-3| + 4\).
04

Part (b) - Calculate \(h(x)\) when \(f(x)=3x-5\) and \(g(x)=-x+2\)

Given: \(f(x) = 3x - 5\) and \(g(x) = -x + 2\). Then, \(h(x) = f(x) + g(x) = (3x - 5) + (-x + 2) = 3x - 5 - x + 2 = 2x - 3\).
05

Part (c) - Calculate \(h(x)\) when \(f(x)=x^2 + 2x\) and \(g(x)=x^2 + x + 2\)

Given: \(f(x) = x^2 + 2x\) and \(g(x) = x^2 + x + 2\). Then, \(h(x) = f(x) + g(x) = (x^2 + 2x) + (x^2 + x + 2) = x^2 + 2x + x^2 + x + 2 = 2x^2 + 3x + 2\).
06

Part (d) - Calculate \(h(x)\) when \(f(x)=-x-5\) and \(g(x)=(x+3)^2\)

Given: \(f(x) = -x - 5\) and \(g(x) = (x + 3)^2\). Then, \(h(x) = f(x) + g(x) = (-x - 5) + (x+3)^2 = -x - 5 + (x^2 + 6x + 9) = x^2 + 5x + 4\).

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Key Concepts

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

Absolute Value
The absolute value of a number is its distance from zero on the number line. It is always non-negative. When you see \(|x-3|\), it means you are finding the distance between \(x\) and 3 without considering direction. The function \(f(x)=|x-3|\) translates to:
  • For \(x \geq 3\), \(f(x) = x - 3\)
  • For \(x < 3\), \(f(x) = 3 - x\)
In the exercise, given \(g(x) = 4\), we have: \[ h(x) = f(x) + g(x) = |x-3| + 4 \] Consequently, we have different expressions for \(h(x)\) based on whether \(x\) is greater or less than 3.
Knowing how to handle absolute values is important when combining functions because they can change based on the input value.
Linear Functions
A linear function is any function that can be graphically represented as a straight line. The general form of a linear function is \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In part (b) of the exercise, we have:
  • \(f(x) = 3x - 5\)
  • \(g(x) = -x + 2\)
Combining these two gives: \[ h(x) = f(x) + g(x) = (3x - 5) + (-x + 2) = 2x - 3 \] Notice how the coefficients of \(x\) and the constants are combined separately.
Linear functions are straightforward, but combining them can alter slopes and intercepts, leading to a different straight line.
Quadratic Functions
A quadratic function is a function where the highest degree of the variable is squared, typically in the form \(f(x) = ax^2 + bx + c\). In part (c) of the exercise, the functions are:
  • \(f(x) = x^2 + 2x\)
  • \(g(x) = x^2 + x + 2\)
Adding these results in: \[ h(x) = f(x) + g(x) = (x^2 + 2x) + (x^2 + x + 2) = 2x^2 + 3x + 2 \] Quadratic functions, when added, combine their respective quadratic, linear, and constant terms.
These functions form parabolas which can open upward or downward and combining them can shift or stretch their shapes.
Combining Functions
Combining functions involves adding their outputs for the same input. This is also known as function addition. For any two functions \(f(x)\) and \(g(x)\), the combined function \(h(x)\) is given by: \[ h(x) = f(x) + g(x) \] In the exercise, we've combined various forms of functions:
  • Absolute value and a constant
  • Two linear functions
  • Two quadratic functions
  • A linear and a quadratic function
For example, given \(f(x) = -x - 5\) and \(g(x) = (x + 3)^2\) in part (d), we calculate: \[ h(x) = (-x - 5) + (x^2 + 6x + 9) = x^2 + 5x + 4 \] Through these examples, we see that combining functions shifts and transforms graphs in different ways.
Understanding combining functions helps in many real-world applications where different factors are mathematically modeled together.

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Most popular questions from this chapter

Given \(f(x)=\sqrt{x}\) and \(g(x)=x-1,\) sketch the graph of each composite function. Then, determine the domain and range of each composite function. a) \(y=f(g(x))\) b) \(y=g(f(x))\)

Use the functions \(f(x)=3 x, g(x)=x-7\), and \(h(x)=x^{2}\) to determine each of the following. a) \((f \circ g \circ h)(x)\). b) \(g(f(h(x)))\) c) \(f(h(g(x)))\) d) \((h \circ g \circ f)(x)\)

Two waves are generated in a ripple tank. Suppose the height, in centimetres, above the surface of the water, of the waves can be modelled by \(f(x)=\sin x\) and \(g(x)=3 \sin x,\) where \(x\) is in radians. a) Graph \(f(x)\) and \(g(x)\) on the same set of coordinate axes. b) Use your graph to sketch the graph of \(h(x)=(f+g)(x)\) c) What is the maximum height of the resultant wave?

Let \(f(x)=\sin x\) and \(g(x)=\cos x\) a) Write an expression as a quotient of functions that is equivalent to tan \(x .\) b) Write an expression as a product of functions that is equivalent to \(1-\cos ^{2} x\) c) Use graphing technology to verify your answers to parts a) and b).

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