Math Problem Statement

Consider the following problem with functions f(x), h(x), and g(x). Part (a) asks for the sum (f + h)(0), and Part (b) asks for the result of (h / g)(-1).

Solution

Let's go through the problem step by step:

Part (a): (f+h)(0)(f + h)(0)

You need to find (f+h)(0)(f + h)(0), which means adding the values of the functions f(x)f(x) and h(x)h(x) at x=0x = 0.

  • From the graph of f(x)f(x), you can see that f(0)f(0) corresponds to the point on the graph where x=0x = 0. By looking at the graph, we can estimate that f(0)=2f(0) = -2.

  • From the table, h(0)h(0) is given directly as h(0)=2h(0) = 2.

Now, calculate: (f+h)(0)=f(0)+h(0)=2+2=0(f + h)(0) = f(0) + h(0) = -2 + 2 = 0

Part (b): (hg)(1)\left(\frac{h}{g}\right)(-1)

This expression requires you to divide h(1)h(-1) by g(1)g(-1).

  • From the table, h(1)=0.5h(-1) = 0.5.
  • The function g(x)=1xx2g(x) = 1 - x - x^2 is provided. To find g(1)g(-1), substitute x=1x = -1 into g(x)g(x): g(1)=1(1)(1)2=1+11=1g(-1) = 1 - (-1) - (-1)^2 = 1 + 1 - 1 = 1

Now, calculate: (hg)(1)=h(1)g(1)=0.51=0.5\left(\frac{h}{g}\right)(-1) = \frac{h(-1)}{g(-1)} = \frac{0.5}{1} = 0.5

Final answers:

(a) (f+h)(0)=0(f + h)(0) = 0
(b) (hg)(1)=0.5\left(\frac{h}{g}\right)(-1) = 0.5

Would you like more details on any step?

Here are 5 questions to expand your understanding:

  1. How would you evaluate (fh)(0)(f - h)(0)?
  2. What is g(0)g(0) and how does it affect other calculations?
  3. How would you graph the function g(x)g(x)?
  4. What is the domain and range of the function f(x)f(x) based on the graph?
  5. How would the solution change if f(x)f(x) was modified slightly?

Tip: When combining functions, always check the values carefully from graphs, tables, or equations!

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Math Problem Analysis

Mathematical Concepts

Function Operations
Graph Interpretation
Substitution in Functions

Formulas

g(x) = 1 - x - x^2
f(x) from graph
h(x) from table

Theorems

Basic Function Operations

Suitable Grade Level

Grades 9-12