Find the Vertical and Horizontal Intercept(s) f(x)=9^(x+1)-4

$f (x) = 9^{x + 1} - 4$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 9^{x + 1} - 4$

Solve the equation.

Rewrite the equation as $9^{x + 1} - 4 = 0$ .

$9^{x + 1} - 4 = 0$

Add $4$ to both sides of the equation.

$9^{x + 1} = 4$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (9^{x + 1}) = \ln (4)$

Expand $\ln (9^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (9) = \ln (4)$

Simplify the left side.

Simplify $(x + 1) \ln (9)$ .

Apply the distributive property.

$x \ln (9) + 1 \ln (9) = \ln (4)$

Multiply $\ln (9)$ by $1$ .

$x \ln (9) + \ln (9) = \ln (4)$

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (9) + \ln (9) - \ln (4) = 0$

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (9) + \ln (\frac{9}{4}) = 0$

Subtract $\ln (\frac{9}{4})$ from both sides of the equation.

$x \ln (9) = - \ln (\frac{9}{4})$

Divide each term in $x \ln (9) = - \ln (\frac{9}{4})$ by $\ln (9)$ and simplify.

Divide each term in $x \ln (9) = - \ln (\frac{9}{4})$ by $\ln (9)$ .

$\frac{x \ln (9)}{\ln (9)} = \frac{- \ln (\frac{9}{4})}{\ln (9)}$

Simplify the left side.

Cancel the common factor of $\ln (9)$ .

Cancel the common factor.

$\frac{x \ln (9)}{\ln (9)} = \frac{- \ln (\frac{9}{4})}{\ln (9)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (\frac{9}{4})}{\ln (9)}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{\ln (\frac{9}{4})}{\ln (9)}$

x-intercept(s) in point form.

x-intercept(s): $(- \frac{\ln (\frac{9}{4})}{\ln (9)}, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 9^{(0) + 1} - 4$

Simplify $9^{(0) + 1} - 4$ .

Simplify each term.

Add $0$ and $1$ .

$y = 9^{1} - 4$

Evaluate the exponent.

$y = 9 - 4$

Subtract $4$ from $9$ .

$y = 5$

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{\ln (\frac{9}{4})}{\ln (9)}, 0)$

y-intercept(s): $(0, 5)$

Step 4

Find the Vertical and Horizontal Intercept(s) f(x)=9^(x+1)-4

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