Question

The function k(x) = (g x h)(x) is graphed below, where g is an exponential function and h is a linear function. If h(x) = x + 1, which option below give the formula for g? h(^x) = 2^x
h(^x) = –2^x
h(^x) = –2^–x
h(^x) = 2^–x

Answer 1

Answer:  The correct option is (D) $g(x)=2^{-x}.$

Step-by-step explanation:  Given that the function k(x) = (g x h)(x) is graphed in the figure, where g is an exponential function and h is a linear function.

We are to find the formula for g, if h(x) = x + 1.

From the graph, we note the following two values :

$k(0)=1,~~k(3)=0.5.$

Now, will check our options one by one.

Option (A) :

Here, $g(x)=2^x.$

S0,

$k(x)=(g\times h)(x)=g(x)\times h(x)=2^x(x+1).$

At x = 0 and 3, we get

$k(0)=2^0(0+1)=1,\\\\k(3)=2^3(3+1)=32\neq 0.5.$

This option is not correct.

Option (B) :

Here, $g(x)=-2^x.$

S0,

$k(x)=(g\times h)(x)=g(x)\times h(x)=-2^x(x+1).$

At x = 0 and 3, we get

$k(0)=-2^0(0+1)=-1\neq 1,\\\\k(3)=-2^3(3+1)=-32\neq 0.5.$

This option is not correct.

Option (C) :

Here, $g(x)=-2^{-x}.$

So,

$k(x)=(g\times h)(x)=g(x)\times h(x)=-2^{-x}(x+1).$

At x = 0 and 3, we get

$k(0)=-2^{-0}(0+1)=-1\neq 1,\\\\k(3)=-2^{-3}(3+1)=-\dfrac{1}{8}(4)=-0.5\neq 0.5.$

This option is not correct.

Option (D) :

Here, $g(x)=2^{-x}.$

S0,

$k(x)=(g\times h)(x)=g(x)\times h(x)=2^{-x}(x+1).$

At x = 0 and 3, we get

$k(0)=2^{-0}(0+1)=1,\\\\k(3)=2^{-3}(3+1)=\dfrac{1}{8}(4)=0.5.$

Therefore, $g(x)=2^{-x}.$

This option is CORRECT.

Hence, (D) is the correct option.

Answer 2

h(x)=-2^-x, yeah this is definitely right