Question

If \(f(x)=2 x^{2}+5\) and \(g(x)=3 x+a,\) find \(a\) so that the \(y\) -intercept of \(f \circ g\) is 23 .

Short answer

The value of 'a' is \( \pm 3 \).

Step-by-step solution

Understand the problem

We need to find the constant 'a' in the function \( g(x) = 3x + a \) such that the y-intercept of the composite function \( f \circ g \) equals 23.

Find the composite function

To find \( f \circ g \), we substitute \( g(x) \) into \( f(x) \). Therefore, \( f(g(x)) = f(3x + a) \).

Substitute \( g(x) \) into \( f(x) \)

Given \( f(x) = 2x^2 + 5 \), substitute \( g(x) = 3x + a \) into \( f(x) \). So, \( f(g(x)) = 2(3x + a)^2 + 5 \).

Simplify the composite function

Expand and simplify \( f(g(x)) = 2(3x + a)^2 + 5 = 2(9x^2 + 6ax + a^2) + 5 = 18x^2 + 12ax + 2a^2 + 5 \).

Determine the y-intercept

The y-intercept occurs when \( x = 0 \). Plugging in \( x = 0 \) into the composite function gives \( f(g(0)) = 18(0)^2 + 12a(0) + 2a^2 + 5 = 2a^2 + 5 \).

Set up the equation

We are given that the y-intercept is 23. Therefore, set up the equation \( 2a^2 + 5 = 23 \).

Solve for 'a'

First, isolate \( 2a^2 \) by subtracting 5 from both sides: \( 2a^2 = 18 \). Then, divide by 2: \( a^2 = 9 \). Finally, take the square root of both sides to get \( a = \pm 3 \).

Key concepts

y-intercept

The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when the independent variable, often denoted by x, is equal to zero. To find the y-intercept, we simply substitute x = 0 into the function and solve for the y-value.
Considering the given composite function from the exercise, we note that the y-intercept of f(g(x)) is where x = 0. Thus, for any function h(x), the y-intercept is found by calculating h(0).

In the context of this exercise, after simplifying the composite function to get 2a² + 5, we substitute x = 0 and this simplifies the expression to 2a² + 5 where x = 0.

function composition

Function composition involves combining two functions where the output of one function becomes the input of another. This is denoted as (f ∘ g)(x), which means f(g(x)).

In the exercise, we are given f(x) = 2x² + 5 and g(x) = 3x + a. To find the composite function f(g(x)), we substitute g(x) into f(x).
So, we replace x in f(x) with g(x), leading to f(g(x)) = f(3x + a).
This results in f(g(x)) = 2(3x + a)² + 5.

Therefore, understanding how to compose functions is key to solving problems where two or more functions need to be combined.

solving quadratic equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0. Solving these equations involves finding the values of x that satisfy the equation.

In the exercise, we simplified the composite function to find the y-intercept and then solved the equation 2a² + 5 = 23.

To isolate a, we first subtracted 5 from both sides resulting in 2a² = 18.
Then, we divided both sides by 2, yielding a² = 9.
Finally, we took the square root of both sides to find the solutions for a, giving us a = ±3.

Thus, solving quadratic equations helps us find critical values needed in various mathematical contexts, including function compositions and y-intercepts.