The function is defined by , where is a constant. If , what is the value of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 37 | Free SAT Prep | Aniko

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Question 37·Easy·Nonlinear Functions

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The function $g$ is defined by $g(x) = k \cdot 4^{x}$ , where $k$ is a constant. If $g(1) = 12$ , what is the value of $k$ ?

When a function with an unknown constant is given, and you are told the function’s value at a specific input, plug that input into the function rule to form an equation in the unknown constant. Simplify any exponents, then solve the resulting one-step linear equation (usually by dividing). Finally, match your numeric result to the correct answer choice, watching out for common mistakes like using the wrong exponent or dividing by the wrong number.

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Use the definition of the function

Start by writing $g(1)$ using the formula $g(x) = k\cdot 4^x$ . What does $g(1)$ equal in terms of $k$ ?

Set up an equation using the given function value

You are told $g(1) = 12$ . Set your expression for $g(1)$ equal to 12 to create an equation involving $k$ .

Isolate k

Once you have an equation of the form $12 = 4k$ , what operation will isolate $k$ on one side of the equation?

Desmos Guide

Write the equation for k

From the function and the given information, you should get an equation of the form $12 = 4k$ . Rearrange this to express $k$ as a fraction (for example, $k = 12/4$ ).

Evaluate the fraction in Desmos

Type 12/4 into Desmos. The numerical value that Desmos shows is the value of $k$ you are solving for.

Step-by-step Explanation

Substitute the given input into the function

You are told that $g(x)=k\cdot 4^x$ and $g(1)=12$ .

Substitute $x=1$ into the function:

g(1)=k\cdot 4^1

But you also know $g(1)=12$ , so set them equal:

12 = k\cdot 4^1.

Evaluate the power and form an equation in k

Now simplify $4^1$ .

Since $4^1 = 4$ , the equation becomes

12 = 4k.

Now you just need to solve this simple linear equation for $k$ .

Solve for k and match the answer choice

Solve $12 = 4k$ by dividing both sides by 4:

k = \frac{12}{4}.

Simplify the fraction:

k = 3.

So the value of $k$ is $3$ , which corresponds to choice C.

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Step-by-step Explanation

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Step-by-step Explanation