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Chapter 7: Problem 22

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ f(10)-g(10) $$

Short Answer

Expert verified
33

Step by step solution

01

- Calculate f(10)

First, evaluate the function f at x = 10 using the given function: \[ f(x) = -3x + 4 \] Substitute x = 10: \[ f(10) = -3(10) + 4 = -30 + 4 = -26 \]
02

- Calculate g(10)

Next, evaluate the function g at x = 10 using the given function: \[ g(x) = -x^{2} + 4x + 1 \] Substitute x = 10: \[ g(10) = -(10)^2 + 4(10) + 1 = -100 + 40 + 1 = -59 \]
03

- Find f(10) - g(10)

Now that we have both f(10) and g(10), subtract g(10) from f(10): \[ f(10) - g(10) = -26 - (-59) \] Simplifying, we get: \[ -26 + 59 = 33 \]

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Key Concepts

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

Function Evaluation
In algebra, functions are mathematical expressions that relate an input to an output. When we talk about 'function evaluation,' we mean the process of finding the output of a function for a specific input. Given a function, you can evaluate it by plugging in different values for the variable.

For example, if you have the function \(f(x) = -3x + 4\), to evaluate it for \(x = 10\), you simply substitute 10 for \(x\) and perform the arithmetic:

  • \( f(10) = -3(10) + 4\)

  • Which simplifies to \( f(10) = -30 + 4 = -26 \)

Similarly, for another function like \( g(x) = -x^2 + 4x + 1 \), evaluating it at \(x = 10\) involves substituting 10 for \(x\) and performing the calculations:
  • \( g(10) = -(10)^2 + 4(10) + 1\)

  • Which simplifies to \( g(10) = -100 + 40 + 1 = -59 \).
Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, involving only non-negative integer exponents. They can have different degrees, determined by the highest power of the variable.

For instance, the functions in the given exercise are polynomial functions:

  • The function \( f(x) = -3x + 4 \) is a linear polynomial because its highest exponent is 1.

  • The function \( g(x) = -x^2 + 4x + 1 \) is a quadratic polynomial because its highest exponent is 2.

Polynomial functions are generally very predictable and easy to work with because they follow simple rules of arithmetic and calculus. In our exercise, we need to evaluate these polynomials at specific values to find the result.
Function Subtraction
Function subtraction involves finding the difference between two function values for the same input. After evaluating each function separately, you subtract the second function's value from the first.

In our example, let's say we have already evaluated \(f(10)\) and \(g(10)\):

  • \( f(10) = -26 \)

  • \( g(10) = -59 \)

To find \( f(10) - g(10) \), you simply subtract:
  • \( f(10) - g(10) = -26 - (-59)\)

  • This simplifies to \( -26 + 59 = 33 \)

This result tells us the output difference between the two functions when the input is 10.

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Most popular questions from this chapter

Find the slope of the line through each pair of points.\(\left(\text {Hint: } \frac{\frac{a}{b}}{\frac{s}{2}}=\frac{a}{b} \div \frac{c}{d}\right)\). $$\left(\frac{1}{6}, \frac{1}{2}\right) \text { and }\left(\frac{5}{6}, \frac{9}{2}\right)$$

For Exercises \(47-58,(a)\) find the slope of the line through each pair of points, if possible, and (b) based on the slope, indicate whether the line through the points rises from left to right, falls from left to right, is horizontal, or is vertical. See Example 6 and FlGURE \(19 .\) $$(-4,1) \text { and }(2,6)$$

Solve each problem. \(\cdot p\) varies jointly as \(q\) and \(r^{2},\) and \(p=200\) when \(q=2\) and \(r=3 .\) Find \(p\) when \(q=5\) and \(r=2\)

Write each formula using the "language" of variation. For example, the formula for the circumference of a circle, \(C=2 \pi r,\) can be written as "The circumference of a circle varies directly as the length of its radius." \(d=2 r,\) where \(d\) is the diameter of a circle with radius \(r\)

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ g(e) $$
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