Problem 35 - Algebra 2

Question

If \(f(x)=4 x, g(x)=2 x-1,\) and \(h(x)=x^{2}+1,\) find each value. $$ h[g(4)] $$

Step-by-Step Solution

Answer

\( h[g(4)] = 50 \)

Step 1: Calculate g(4)

First, let's find the value of \( g(4) \) using the function \( g(x) = 2x - 1 \). Substitute \( x = 4 \) into the equation: \[ g(4) = 2(4) - 1 = 8 - 1 = 7 \]. Thus, \( g(4) = 7 \).

Step 2: Evaluate h[g(4)]

Now that we know \( g(4) = 7 \), we need to evaluate \( h(7) \) using the function \( h(x) = x^2 + 1 \). Substitute \( x = 7 \) into the equation: \[ h(7) = 7^2 + 1 = 49 + 1 = 50 \]. Thus, \( h[g(4)] = 50 \).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the output of a function given a specific input. In a way, it's like asking the function a question: "What is your response if I give you this input?" When evaluating a function, you substitute the input value into the function's expression in place of the variable, usually denoted as \( x \).
For example, if we have a function \( f(x) = 4x \), to find \( f(4) \), we substitute 4 for \( x \) in the equation. This results in:

• \( f(4) = 4 \times 4 = 16 \)

Why is function evaluation important?

• It helps us understand how functions behave when different values are fed into them.
• It is a critical step in solving more complex problems involving multiple functions, such as composite functions.

In composite functions, like the one we solved in the exercise, you first evaluate an inner function, and then use that result to evaluate an outer function.

Quadratic Function

A quadratic function is a type of mathematical expression that has the general form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This means the highest power of \( x \) in a quadratic function is 2. In the exercise, we see a specific quadratic function: \( h(x) = x^2 + 1 \).
Quadratic functions create parabolic graphs, which are U-shaped curves that can open upwards or downwards, depending on the sign of \( a \):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Characteristics of Quadratic Functions:

• The vertex is the highest or lowest point of the parabola.
• The axis of symmetry is a vertical line that passes through the vertex, giving the parabola its symmetrical shape.
• The roots or zeros are the points where the parabola crosses the x-axis.

These features make quadratic functions extremely useful in various fields like physics, engineering, and economics, often modeling things such as projectile motion or maximizing profits.

Linear Function

A linear function is the simplest type of function and has a form of \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The graph of a linear function is a straight line. In our exercise, \( g(x) = 2x - 1 \) is a linear function.
Slope and Y-Intercept:

• The slope \( m \) measures how steep the line is. If \( m = 2 \), like in our exercise, it means that for every increase of 1 unit along the x-axis, the y value increases by 2 units.
• The y-intercept \( b \) is the point where the line crosses the y-axis. Here, \( b = -1 \), meaning the line crosses the y-axis at \( (0, -1) \).

Linear functions are fundamental in algebra and everyday situations. They describe relationships where one thing is consistently a certain factor of another, such as speed (distance over time) or converting currencies at a constant rate. Understanding linear functions helps in grasping how changing one variable impacts another linearly.