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Chapter 3: Problem 55

If \(f(x)=3 x+3, g(x)=4 x^{2}-6 x+3,\) and \(h(x)=5 x^{2}-7\) find the following. $$ h(-3) $$

Short Answer

Expert verified
38

Step by step solution

01

Identify the function

First, determine which function is needed to solve this exercise. Since the question asks for \( h(-3) \), we will use the function \( h(x) = 5x^2 - 7 \).
02

Substitute -3 into the function

Replace \( x \) with \(-3\) in the function \( h(x) = 5x^2 - 7 \). This gives us \( h(-3) = 5(-3)^2 - 7 \).
03

Calculate \(-3^2\)

Calculate \((-3)^2\), which is \( 9 \).
04

Multiply by 5

Multiply the result from Step 3 by 5. So, \( 5 imes 9 = 45 \).
05

Subtract 7

Subtract 7 from the result of Step 4. So, \( 45 - 7 = 38 \).

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Key Concepts

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

Substitution in Functions
Substitution in functions is a fundamental concept in algebra that involves replacing the variable in a function with a specific number or expression. It's like filling in a blank with a specific value to see what the outcome will be. In this case, we need to find the value of the function for a given input. For example, when evaluating the function \( h(x) = 5x^2 - 7 \) for \( x = -3 \), you simply replace every "\( x \)" in the function with "\( -3 \)". This substitution gives us \( h(-3) = 5(-3)^2 - 7 \). A few key points to remember while doing substitution are:
  • Replace every occurrence of the variable with the given value.
  • Be cautious of signs, especially with negative values and exponents.
  • Follow the order of operations: parentheses, exponents, multiplication, and addition/subtraction.
This step translates the abstract function into an equation you can solve for a specific outcome.
Function Notation
Function notation is a way to name a function and indicate the input value. It鈥檚 commonly written as \( f(x) \). The "\( f \)" stands for the function's name, while the "\( x \)" inside the parentheses represents the variable or input. This form clearly conveys the function's rule in terms of its variable.
For example, in our problem, we have three functions: \( f(x) = 3x + 3 \), \( g(x) = 4x^2 - 6x + 3 \), and \( h(x) = 5x^2 - 7 \). Here, "\( h(x) \)" represents the function that we are working with. In this case:
  • "\( h \)" is the name of the function, helping to differentiate it from other functions like \( f \) and \( g \).
  • "\( x \)" is the variable input, which can be replaced by any number.
  • When you see something like \( h(-3) \), it specifies that the input value is \( -3 \).
Function notation not only helps organize different equations but also makes it clear what input is being used to find the corresponding output.
Quadratic Functions
Quadratic functions are a type of polynomial function that typically feature a squared term and take the standard form \( ax^2 + bx + c \). These functions create a parabolic shape when graphed, which can open upwards or downwards depending on the sign of the leading coefficient. In the given exercise, the function \( h(x) = 5x^2 - 7 \) is a quadratic function.
Quadratic functions are characterized by:
  • The degree of the polynomial being two, with the highest exponent on the \( x \) term being 2.
  • Having a curve known as a parabola whose direction depends on the coefficient \( a \). Here, \( a = 5 \), which is positive, meaning it opens upwards.
  • The vertex of the parabola, which for simple forms like \( ax^2 + c \), is at the point \((0, c)\).
In the specific example of \( h(x) \), replacing \( x \) with \(-3\) lets us explore the value of the function at this point on the parabola. Quadratic functions are key to understanding more complex algebraic concepts and appear in various real-world applications.

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Most popular questions from this chapter

Think about the appearance of each graph. Without graphing, determine which equations represent functions. Explain each answer. See the Concept Check in this section. $$ x+y=-5 $$

Describe a function whose domain is the set of people in your hometown.

Explain why it is a good idea to use three points to graph a linear equation.

Explain when a dashed boundary line should be used in the graph of an inequality.

Given the following functions, find the indicated values. \(h(x)=-x^{2}\) a. \(h(-5)\) b. \(h\left(-\frac{1}{3}\right)\) c. \(h\left(\frac{1}{3}\right)\)
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