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Chapter 9: Problem 16

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1\) \(f\left(\frac{7}{3}\right)\)

Short Answer

Expert verified
The value of \(f\left(\frac{7}{3}\right)\) is -3.

Step by step solution

01

Understand the Given Functions

Given are two functions: \(f(x) = -3x + 4\) and \(g(x) = -x^2 + 4x + 1\). We need to focus on the function \(f(x)\) for this exercise.
02

Substitute the Given Value for x

We need to find \(f\left(\frac{7}{3}\right)\). Substitute \(x = \frac{7}{3}\) into the function \(f(x)\).
03

Perform the Substitution

Substitute \(x\) in \(f(x) = -3x + 4\) with \(\frac{7}{3}\): \[f\left(\frac{7}{3}\right) = -3 \left(\frac{7}{3}\right) + 4\]
04

Simplify the Expression

Simplify the expression: \[-3 \left(\frac{7}{3}\right) = -7\] So, \[f\left(\frac{7}{3}\right) = -7 + 4 = -3\]

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Key Concepts

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

Linear Functions
Linear functions are fundamental in algebra and are typically expressed in the form of \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) represents the y-intercept. Their graph is a straight line, making them relatively simple to understand and work with. In our given exercise, the function \(f(x) = -3x + 4\) is a linear function:
  • The slope \(m\) is -3, indicating how steep the line is.
  • The y-intercept \(b\) is 4, showing where the line crosses the y-axis.
Understanding these characteristics is crucial because it helps in interpreting the equation and predicting the behavior of the function when different values are substituted for \(x\).
Substitution Method
The substitution method is a straightforward way to evaluate functions. It involves replacing the variable, \(x\) in this case, with a given value and simplifying the resulting expression.
In our exercise, we're asked to find \(f\bigg(\frac{7}{3}\bigg)\). Here's how:
  • First, we identify the function we are working with, which is \(f(x) = -3x + 4\).
  • Next, we substitute \(x = \frac{7}{3}\) into the function, resulting in the expression \(-3\bigg(\frac{7}{3}\bigg) + 4\).
After substituting, our task is to simplify this expression to find the value of the function at the given point.
Algebraic Simplification
Algebraic simplification is the process of making an algebraic expression as simple as possible. This includes combining like terms, performing arithmetic operations, and reducing fractions.
In our given problem, we simplify the expression \(-3\bigg(\frac{7}{3}\bigg) + 4\) as follows:
  • First, we multiply \(-3\) by \(\frac{7}{3}\). The \(3\)s cancel out, leaving us with \(-7\).
  • Next, we add 4 to \(-7\), resulting in \(-3\).
Thus, \(f\bigg(\frac{7}{3}\bigg) = -3\). Simplification is a vital skill in algebra as it allows us to rewrite complex expressions in a more manageable form, making it easier to understand and solve problems.

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Most popular questions from this chapter

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ g)\left(-\frac{1}{2}\right) $$

Determine whether each relation defines y as a function of \(x .\) (Solve for y first if necessary.) Give the domain. $$ y=2 x-6 $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ g)(4) $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (f \circ g)(0) $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{f}{h}\right)(1) $$
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