Problem 58 Find the function values. \(f(... [FREE SOLUTION]

Chapter 15: Problem 58

Find the function values. \(f(t)=6-t^{2}\) a) \(f(-1)\) b) \(f(1)\) c) \(f(3 a)\)

Short Answer

Expert verified

a) \( f(-1) = 5 \), b) \( f(1) = 5 \), c) \( f(3a) = 6 - 9a^2 \)

Step by step solution

Understand the problem

The given function is \( f(t) = 6 - t^2 \). The task is to find the value of the function for specific values of \( t \): \( t = -1 \), \( t = 1 \), and \( t = 3a \).

Calculate \( f(-1) \)

Substitute \( t = -1 \) into the function: \[ f(-1) = 6 - (-1)^2 \ f(-1) = 6 - 1 \ f(-1) = 5 \]

Calculate \( f(1) \)

Substitute \( t = 1 \) into the function: \[ f(1) = 6 - 1^2 \ f(1) = 6 - 1 \ f(1) = 5 \]

Calculate \( f(3a) \)

Substitute \( t = 3a \) into the function: \[ f(3a) = 6 - (3a)^2 \ f(3a) = 6 - 9a^2 \]

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Key Concepts

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

quadratic functions

Quadratic functions are mathematical expressions where the highest power of the variable is squared. Generally, they take the form: f(x) = ax^2 + bx + c where:

a, b, and c are constants
x is the variable

In a quadratic function, the graph is a parabola, which can open upwards or downwards. The function in our problem is: f(t) = 6 - t^2 This is a specific type of quadratic function where:

a = -1
b = 0
c = 6

Since the coefficient of t^2 is negative, the parabola opens downwards.

substitution method

The substitution method is a technique used to find the values of functions. This involves replacing the variable in the function with a specific value, then simplifying the resulting expression. In our exercise, we need to substitute given values of t into the function:

First, we replace t with -1 to find f(-1)
Next, we replace t with 1 to find f(1)
Finally, we replace t with 3a to find f(3a)

Through this process, we manage to transform the function value problem into simple arithmetic. Let's look at these steps: f(-1)=6-(-1)^2=6-1=5 f(1)=6-1^2=6-1=5 f(3a)=6-(3a)^2=6-9a^2

algebraic expressions

Algebraic expressions combine numbers, variables, and arithmetic operations. In function problems, these expressions must be simplified to find specific function values. Let's take our function: f(t)=6-t^2
Here, we perform operations like addition, subtraction, and squaring based on the given values for t. Simplifying algebraic expressions requires understanding order of operations (PEMDAS/BODMAS):

P/B: Parentheses/Brackets
E/O: Exponents/Orders (powers and roots)
MD/DM: Multiplication and Division
AS: Addition and Subtraction

For example: Replacing t with -1: f(-1)=6-(-1)^2=6-1=5 Replacing t with 1: f(1)=6-1^2=6-1=5 Replacing t with 3a: f(3a)=6-(3a)^2=6-9a^2 These examples illustrate the importance of following correct algebraic steps to arrive at accurate solutions.

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91影视

Find the function values. \(f(t)=6-t^{2}\) a) \(f(-1)\) b) \(f(1)\) c) \(f(3 a)\)

Short Answer

Step by step solution

Understand the problem

Calculate \( f(-1) \)

Calculate \( f(1) \)

Calculate \( f(3a) \)

Key Concepts

quadratic functions

substitution method

algebraic expressions

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