Question 174·Medium·Math

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The function $h$ is defined by

h(x)=k\,b^{x-2}+7,

where $k$ and $b$ are positive constants. If $h(2)=12$ and $h(4)=27$ , what is the value of $h(5)$ ?

For exponential-function questions with unknown constants, first plug in the given input–output pairs to create simple equations. Choose the input that makes the exponent easiest (often giving an exponent of 0 or 1) to solve for one constant, then use the other input to find the remaining constant. Once both parameters are known, carefully substitute the requested $x$ -value, pay attention to the exponent, and perform the arithmetic step by step to avoid small calculation errors.

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Start with the easier input

Look at $h(2)$ . What happens to the exponent $x-2$ when $x=2$ ? Use that to write a simple equation for $k$ .

Find the base $b$

Once you know $k$ , substitute $x=4$ into $h(x)$ to create an equation involving $b^{2}$ . Solve this equation to find $b$ .

Now plug in $x=5$

After you have both $k$ and $b$ , substitute $x=5$ into $h(x)=k b^{x-2}+7$ . Be careful with the exponent $x-2$ and with the final $+7$ .

Desmos Guide

Compute $k$

In Desmos, type 12-7 to verify that $h(2)-7=5$ , so $k=5$ because $h(2)=k+7$ .

Compute $b$

Type 27-7 to get $20$ , then type 20/5 to confirm $b^2=4$ . Finally, type sqrt(4) and note the positive result for $b$ (since $b$ is given as positive).

Evaluate $h(5)$

In Desmos, enter the expression 5*2^3+7 (using the $k$ and $b$ you found). The output is the value of $h(5)$ ; read that value directly from Desmos.

Step-by-step Explanation

Use $h(2)=12$ to find $k$

Substitute $x=2$ into the function

h(x)=k b^{x-2} + 7

When $x=2$ , the exponent becomes $2-2=0$ , so $b^{0}=1$ :

h(2)=k\bigl(b^{0}\bigr)+7 = k+7

We are told $h(2)=12$ , so

k+7=12 \Rightarrow k=5

Now we know $k=5$ .

Use $h(4)=27$ to find $b$

Substitute $x=4$ and $k=5$ into the function:

h(4)=5 b^{4-2} + 7 = 5b^{2} + 7

We are told $h(4)=27$ , so

5b^{2} + 7 = 27 \Rightarrow 5b^{2} = 20 \Rightarrow b^{2}=4

Because $b$ is positive, $b=2$ .

Evaluate $h(5)$ using $k$ and $b$

Now substitute $x=5$ , $k=5$ , and $b=2$ into the function:

h(5)=5 b^{5-2} + 7 = 5 b^{3} + 7

Since $b=2$ , we have $b^{3} = 2^{3}=8$ , so

h(5)=5\cdot 8 + 7 = 40 + 7 = 47

So the value of $h(5)$ is 47.

Question 174·Medium·Math

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation