Let the function be defined by where is a positive constant with . If , what is the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 186 | Free SAT Prep | Aniko

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Question 186·Hard·Nonlinear Functions

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Let the function $f$ be defined by

f(x) = a^{x-1} + a^{1-x}

where $a$ is a positive constant with $a \ne 1$ . If $f(3) = 5$ , what is the value of $f(5)$ ?

For nonlinear function questions like this, focus on rewriting expressions so that the unknown cancels or becomes unnecessary. Here, plug in the given $x$ to express $f(3)$ in terms of a simpler expression like $a^2 + 1/a^2$ , then express the target $f(5)$ as $a^4 + 1/a^4$ and use algebraic identities such as $(u + 1/u)^2 = u^2 + 2 + 1/u^2$ to relate them. This lets you avoid solving for $a$ directly and quickly compute the needed value from the information given.

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Start by plugging in the given $x$ value

Substitute $x=3$ into $f(x) = a^{x-1} + a^{1-x}$ . What does $f(3)$ become in terms of powers of $a$ ?

Look for a pattern in the exponents

For $x=3$ , the exponents become $2$ and $-2$ . For $x=5$ , what do the exponents become? How are $2$ and $4$ related?

Express $f(5)$ using $f(3)$

If you know $a^2 + 1/a^2$ , how can you find $a^4 + 1/a^4$ ? Try squaring $a^2 + 1/a^2$ and simplify the result.

Finish with a simple arithmetic step

After you relate $a^4 + 1/a^4$ to $a^2 + 1/a^2$ , substitute the value you found from $f(3)$ and compute the final result.

Desmos Guide

Define the function with a slider for $a$

In Desmos, type f(x) = a^(x-1) + a^(1-x). Desmos will create a slider for a. Make sure the slider range includes positive values not equal to 1 (for example, 0.1 to 5).

Adjust $a$ so that $f(3)$ equals 5

Add a new line with f(3). Then move the a slider until the numeric value shown for f(3) is as close to 5 as possible. (There may be two such values of a; either one is fine.)

Read off the value of $f(5)$

Once a is adjusted so that f(3) is 5, add a new line with f(5). The number Desmos displays for f(5) is the value you are looking for; match this number to the closest answer choice.

Step-by-step Explanation

Write $f(3)$ in terms of powers of $a$

Substitute $x = 3$ into the definition of $f$ :

f(3) = a^{3-1} + a^{1-3} = a^2 + a^{-2} = a^2 + \frac{1}{a^2}.

We are told $f(3) = 5$ , so we know

a^2 + \frac{1}{a^2} = 5.

Express $f(5)$ using similar powers of $a$

Now substitute $x = 5$ into $f(x)$ :

f(5) = a^{5-1} + a^{1-5} = a^4 + a^{-4} = a^4 + \frac{1}{a^4}.

So our goal is to find $a^4 + 1/a^4$ using the fact that $a^2 + 1/a^2 = 5$ .

Relate $a^4 + 1/a^4$ to $a^2 + 1/a^2$

Square the expression $a^2 + \dfrac{1}{a^2}$ :

\left(a^2 + \frac{1}{a^2}\right)^2 = a^4 + 2 + \frac{1}{a^4}.

The right side can be grouped as

a^4 + \frac{1}{a^4} + 2.

So we have the identity

\left(a^2 + \frac{1}{a^2}\right)^2 = a^4 + \frac{1}{a^4} + 2.

This lets us solve for $a^4 + 1/a^4$ in terms of $a^2 + 1/a^2$ .

Use the identity and the known value to compute $f(5)$

From the identity in the previous step,

a^4 + \frac{1}{a^4} = \left(a^2 + \frac{1}{a^2}\right)^2 - 2.

We know $a^2 + 1/a^2 = 5$ , so

a^4 + \frac{1}{a^4} = 5^2 - 2 = 25 - 2 = 23.

But $f(5) = a^4 + 1/a^4$ , so $f(5) = 23$ , which corresponds to choice C.

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation